The problem with most supplements, especially at megadose levels, is that supplement companies and researchers tend to study only the glamorous positive aspects in the preliminary studies. Thus, in the beginning the health world is flooded with good reports. Then, a decade or two later, some negative aspects of the supplement beging leaking out and this is exactly what is happening with the antioxidants in some cases.
Before I go on, I want to call attention to something important: I know of no study that shows that antioxidants in foods can cause the kind of issues below. In fact, study after study shows the benefits of eating fruits and vegetables in particular, which are almost always packed with antioxidants. If you haven't seen the research, take a look at my page on The Incredible Benefits of Fruit and Vegetable Consumption and I'm sure you'll become a believer.
NEWS FLASH: The co-discoverer of DNA (James Watson) believes that a lack of oxidants (free radicals) and not a lack of antioxidants is what causes the inflammation that kills of your pancreatic beta cells that make insulin.  His argument is that exercise, which creates a flood of free radicals, actually greatly benefits insulin resistance and protects against diabetes. Furthermore, he points out that these benefits go away if you give them antioxidants before the exercise! He did further research and found that these free radicals are actually critical for protein formation and maintenance. See #5 below for the actual study that Dr. Watson was referring to.
1. Grape Seed + Vitamin C. There was a very interesting study done on 69 people with high blood pressure. In this study they had four groups: those on a) grape seed, b) Vitamin C (500 mg/day), c) grape seed extract + Vitamin C and d) neither. Now both the grape seed extract and Vitamin C groups caused very small drops in blood pressure, one statistically significant and the other not. And so the two together combined to produce an even bigger drop, right? Actually, what happened is that the two together produced a pretty significant average increase in blood pressure of 4.8 mm systolic. The authors, with the characteristic understatement of researchers, noted that "these results suggest caution for hypertensive subjects taking supplements containing combinations of vitamin C and polyphenols."  Therefore, it appears that mixing two powerful antioxidants together can at times do the exact opposite of what is hoped for. Seriously, who would have guessed that a well-respected herb and a well-researched antioxidant like Vitamin C could together raise blood pressure.
2. Tempol (SOD mimetic) + Vitamin C. An animal study in 2010 showed that the combination of these two antioxidants actually decreased oxygen delivery to muscles and increased the "fatigability" of the muscles.  So what caused the underlying problems? The researchers involved in the study pointed out that peroxide, a free radical, is actually involved in vasodilation, i.e. relaxig of the blood vessels and arteries. If you get enough antioxidants to actually lower peroxide levels too far, it can have a negative impact. This may explain what was going in #1 above as well.
3. Vitamin E. Vitamin E is considered one of your most important antioxidants. It protects against lipid peroxidation and, therefore, is considered artery protective.  Epidemiological studies have shown the important of Vitamin E's antioxidant superpowers. However, megadosing the alpha form of Vitamin E has turned out to be a very bad idea. And the alpha form is still the one used in the great majority of supplements out there.
The reason is that alpha tocopherol - tocopherol is essentially another name for Vitamin E - reduces two of your body's cancer-fighting forms, delta and gamma tocopherol. One research summary stated it this way: "Our recent results in animal models have shown the cancer preventive activity of Î³- and Î´-tocopherols as well as a naturally occurring mixture of tocopherols, and the lack of cancer preventive activity by Î±-tocopherol."  And this poor formulation of Vitamin E - leaving out the delta and gamma tocopherols - may have actually killed a few people according to one study, a large meta-analysis that found that Vitamin E (along with the antioxidants Vitamin A and Betacarotene) lead to increased mortality.  The issues seem to occur at higher dosages:
"One analysis found an increased risk of death at doses of 400 IU/day, although the risk began to increase at 150 IU. In another analysis of studies of antioxidant supplements for disease prevention, the highest quality trials revealed that vitamin E, administered singly (dose range 10 IUâ€“5,000 IU/day; mean 569 IU) or combined with up to four other antioxidants, significantly increased mortality risk." 
So with Vitamin E, the risks appear to be twofold: a) improperly formulation using only alpha tocopherol and b) megadosing beyond about 150 IU.
4. Vitamin C versus Orange Juice. Orange juice is one of the most maligned of juices. It is fairly sweet and is pasteurized (heated) for bizarre reasons and is considered "unnatural" by many in the health community. (OJ companies pasteurize it for increased shelf life and then actually hire companies to add back in artificial flavors to make up for the taste that is destroyed through heating!) So researchers put these two up head-to-head to see which one would best protect against DNA damage. They designed the study so that both raised plasma Vitamin C levels the same amount. To their surprise, the Vitamin C alone provided no protection whatsoever, but the orange juice provided significant protection. Their conclusion? "The protective effect of BOJ was not explained by vitamin C alone, thus other phytochemicals could be involved." 
Now this last study shows what I consider the subtle danger of megadosing antioxidants. Many men do it thinking they are getting substantial protection, when they are not. And, even more important, they ignore the power of food because a) it's easy to take a pill and b) slick marketing by many of supplement companies.
NOTE: The dosage in the above study was pretty low (150 mg) and some might argue that Vitamin C does have studies showing it a) lowers blood pressure and b) provides cancer and other types of protection. This is true. But the point is that often, at least at high enough dosages, Vitamin C when coupled with another antioxidant can cause negative effects. And, furthermore, isn't it strange that a little orange juice could do what Vitamin C could not. Now imagine if you throw in the another 9 fruits and vegetables that you should be getting each day and you get an idea of the incredible broad range of protection that whole foods can supply.
However, there is a chance that if you combine Vitamin C with other antioxidants that you may actually do more harm than good as study #1 shows above. Remember that Vitamin C is usually studied alone and not in combination with other supplements or antioxidants. So it appeart to be a gamble with our current, very limited knowledge. Furthermore, even if combinations of supplements do not bite you, it is very possible that an incorrect form may as Vitamin E shows.
5. Antioxidants Reduce Insulin Sensitivity Benefits of Exercise! A PNAS study showed that a gram of Vitamin C and 400 IU of Vitamin E reduced the insulin benefits of exercise.  This is an incredible find and in my opinion casts doubt on megadosing of antioxidants. See the News Flash above for further explanation.  The authors summarized with the following sobering thoughts:
"If transient increases in oxidative stress are capable of counteracting insulin resistance in humans, it is possible that preventing the formation of ROS by, for example, antioxidants might actually increase, rather than decrease, the risk of type 2 diabetes. While this remains to be determined, one metaanalysis of previously published studies (27) suggests that high dietary intake of fruits and vegetables, a source of antioxidants but also of numerous other bio-active compounds, may actually decrease the risk for type 2 diabetes. Nevertheless, and as stated by Hamer and Chida (27), all larger intervention trials evaluating the diabetes-preventive potential of defined antioxidant supplements have been unable to find any positive effects of supplementation (28â€“30). Moreover, antioxidant use in type 2 diabetics has been linked to increased prevalence of hypertension (31) and use of antioxidant supplements has recently been proposed to increase overall mortality in the general population (32)."
Again, notice the trend: food helps but megadosing does the opposite (in diabetics at least). This is a common pattern in the studies: supplement provide little benefit or harmful effects and natural, whole foods improve health. I discuss this in greater detail in my link on The Incredible Research-Backed Health Benefits of Fruits and Vegetables.
DISCUSSION: There are GREAT reasons to take antioxidants. In the case of Vitamin C, studies show that it increases intercourse frequency, boosts endothelial function, improves inflammatory cytokines, lowers cortisol reactivity and on and on the list goes. I cover this in my link on Why Take Vitamin C?
Finally, the bottom line is that you can get all the benefits that you are looking for, such as lowered blood pressure, increased nitric oxide, decreased cancer and heart disease risk with food. If you want antioxidants, get them in food along with the 100's of other phytochemicals that you need.
6. N-Acetylcysteine (NAC) May Increase Prostate and Skin Cancer Risk. NAC is currently a very popular supplement and many men take it for liver support and anti-aging purposes. However, a recent study casts doubt on the wisdom of this. Researchers were concerned that a selenium and Vitamin E study had not done as well as anticipated and appeared to actually fuel prostate cancer. (Other study work with selenium has been positive however.) What the researches found in a study on mice was that N-acetylcysteine "decreased ROS levels in Nkx3.1 mutant mouse prostates, it failed to reduce prostatic epithelial hyperplasia/dysplasia. Rather, NAC treatment increased epithelial cell proliferation and promoted the expression of a pro-proliferative gene signature."  Basically, you need ROS (reactive oxygen species or free radical) production for certain transcription factors that fight the spread of cancer. In other words, ROS is just as important as antioxidants and megadosing on antioxidants can disturb that delicate balance. A followup study showed that NAC may increase the metastes of melanoma as well. 
7. EGCG From Green Tea Extract and Liver Damage. EGCG is the most powerful antioxidant in green tea and is responsible for many of it's excellent properties. However, multiple reports and studies now show that it can be liver toxic in higher dosages and some people have actually been injured from it. Researchers are not entirely sure of the mechanism of action, but it is important to note that it can actually lower one of the body's master antioxidants, glutathione, in some tissues. The liver is highly dependent on glutathione for example. In fact, researchers in a recent study wrote that "In contrast, EGCG dose- and time-dependently decreased the amount of intracellular glutathione (GSH) levels." 
NOTE: You may be interested to know that some antioxidants have been shown to boost testosterone (in large doses). See my link on Antioxidants and Your Testosterone for more information.
1) Journal of Hypertension, Feb 2005, 23(2):427-434, "The combination of vitamin C and grape-seed polyphenols increases blood pressure: a randomized, double-blind, placebo-controlled trial"
3) Journal of Applied Physiology, Dec 2008, 105(6):889-1896, "Effects of antioxidants on contracting spinotrapezius muscle microvascular oxygenation and blood flow in aged rats"
4) Experimental Physiology, Sep 2009, 94(9):961â€“971, "The effects of antioxidants on microvascular oxygenation and blood flow in skeletal muscle of young rats"
5) British Journal of Nutrition, 2007, 97:639â€“643, "Orange juice vs vitamin C: effect on hydrogen peroxide-induced DNA damage in mononuclear blood cells"
6) Cardiovascular Research, 57(2):563-571, "Vitamin E inhibits lipid peroxidation-induced adhesion molecule expression in endothelial cells and decreases soluble cell adhesion molecules in healthy subjects"
7) Cancer Prev Res, May 1 2012, 5:701, "Does Vitamin E Prevent or Promote Cancer?"
8) AMA, 2007 Feb 28, 297(8):842-57, "Mortality in randomized trials of antioxidant supplements for primary and secondary prevention: systematic review and meta-analysis"
9) http://ods.od.nih.gov/factsheets/VitaminE-HealthProfessional/, Office of Dietary Supplements, Vitamin E Fact Sheet
11) PNAS, May 26 2009, 106(21), "Antioxidants prevent health-promoting effects of physical exercise in humans"
12) PLoS One, 2012, 7(10):e46792, "Antioxidant treatment promotes prostate epithelial proliferation in Nkx3.1 mutant mice"
13) Nature, Nov 2015, 527(23), "Oxidative stress inhibits distant metastasis by human melanoma cells"
14) Mol Nutr Food Res, 2009 Mar, 53(3):349-60, "The effects of green tea (-)-epigallocatechin-3-gallate on reactive oxygen species in 3T3-L1 preadipocytes and adipocytes depend on the glutathione and 67 kDa laminin receptor pathways"