When I started my testosterone cypionate shots, I found out a few months later that they had a policy that, if your PSA was over 2.5 or you had a rolling one year rise greater than 1.0, you would have to get urological clearance to continue TRT. Of course, that is something I believe should be communicated before you start testosterone therapy, not afterwards! I later wondered if these guidelines were grounded on best practices and resarch or were just something to keep the attorneys happy.
Below I attempt to go into detail on that question but first want to point out that some PSA increase is to be expected when you go on testosterone therapy. This is simply because some of your new found testosterone will be converted to DHT (dihydrotestosterone) through the 5-alpha reductase enzyme. And this increase in DHT will enlarge the prostate a little and increase PSA. One review found that "averaging several investigations of the effect of TRT on PSA, men receiving testosterone will have an associated increase of 0.30 ng/mL/y in serum PSA, with older men experiencing a greater increase of 0.43 ng/mL/y." 
Of course, the question is in the cases where prostate increases exceed the norms. Let me give you some Examples of PSA Levels That Could Make you Quit Testosterone Therapy:
1. PSA Cutoff of 3.0, 4.0 or 5.4 ng/ml. Urologists and endocrinologists have a term called "PSA Cutoff for Biopsy," which is just what it says. Values for this threshold have varied considerably over the years. The initial values started at 4.0 ng/ml from what I have read. However, a followup study in 2005 said that a significant number of lives could be saved by lowering that threshold down to 3.0.  However, this was contradicted by a recent study that recommended that the value be raised actually to 5.4 ng/ml.  One study even concluded that no reasonable PSA value could be select due to the fact that "an optimal PSA level must weigh detection of cancer with discernment of cancer and non-cancerous conditions...The researchers concluded that the optimal PSA level from which to recommend a prostate biopsy remains unclear since no level appeared to provide both high sensitivity and specificity." 
Out of this chaos, your physician will have to select a threshold at which he would pull you off of testosterone therapy. As you can see, it could be just about anything from 2.5 to 5.5, so discuss the subject with him or her ahead of time.
2. Rolling 1-Year PSA Rise Greater Than 1.4-1.5 ng/ml. One well-known set of TRT guidelines recommended that a "rolling PSA Rise > 1.4" would be the threshold of concern.  This is a concept that my HRT clinic used for example. They looked for a rise of 1.0 from year to year and , therefore, were even more strict.
2A. A Rise in PSA Velocity of Greater Than 0.4 - 2.0 ng/ml per Year. PSA Velocity is a concept similar to the rolling data points mentioned in #2. The difference is mostly philosophical: the idea is based on initial study work that showed that rapid rises in PSA tended to precede the more deadly forms of prostate cancer.  Wide variations in guidelines existed as in #2 and a larger, a more recent study concluded essentially that PSA velocity was useless and should not be used in any clinical guidelines.  Again, though, what counts is the number that your physician will use to make you stop your treatments.
CONCLUSION: Various PSA levels have been proposed removing a man from testosterone replacement and/or screening with a biopsy. However, considerable controversy exists over the validity of these levels with the more recent studies showing that they are largely useless. I still value the PSA though and have self-tested for it, because I want to know if I have hidden inflammation. (See my Inexpensive Testosterone Labs for ways to self-test using Labcorp and Sonora Quest to do your testing without a doctor's order at a price that is the typical copay through insurance!) It should also be pointed out that some experts are concerned about "needle tracking," which is the idea that a biopsy needle often punctures a prostate cancer nodule and spreads it. In other words, biopsies could potentially be dangerous.  Some urologists are using MRI's now.
NOTE: I have many other pages on prostate cancer, BPH, etc. here in my page called A Summary Page for Prostate Issues.
This issue is near and dear to my heart, because it is something that did actually happened to me. My testosterone rose rapidly from one 3 month test to the next, something I discuss on my page High PSA But No Prostate Cancer, thus violating about 3 or 4 of these guidelines at one time. And I was yanked off of TRT immediately despite my pleadings to be allowed to stay on testosterone at least until I talked to the urologist. It was a simple "Do Not Pass Go. Go Directly to Jail" card. I was able to get back on my original testosterone protocol about a month later, but I recmmend that you proactively discuss all of this ahead of time with your physician. I was definitely caught flat-footed but won't again.
Rev Urol, 2004; 6(Suppl 6):S41ï¿½S43, "Rising PSA during Testosterone Replacement Therapy"
2) Urology, 2005 Feb, 65(2):343-6, "Is additional testing necessary in men with prostate-specific antigen levels of 1.0 ng/mL or less in a population-based screening setting? (ERSPC, section Rotterdam)"
3) Indian J Med Res, 2014 Jun, 139(6): 851ï¿½856, "Raising cut-off value of prostate specific antigen (PSA) for biopsy in symptomatic men in India to reduce unnecessary biopsy"
4) Journal of the American Medical Association. 2005;294:66-70 as summarized by Cancer Connect, "No Definitive PSA Cutoff for Prostate Biopsies"
5) J Clin Endocrinol Metab, 2010 Jun, 95(6):2536-59, "Testosterone Therapy in Men with Androgen Deficiency Syndromes: An Endocrine Society Clinical Practice Guideline"
6) N Engl J Med, 2004 Ju6) N Engl J Med, 2004 Jul 8, 351(2):125-35, "Preoperative PSA velocity and the risk of death from prostate cancer after radical prostatectomy"
7) J Natl Cancer Inst. 2011 Mar 16;103(6):462-9, "An empirical evaluation of guidelines on prostate-specific antigen velocity in prostate cancer detection"
8) Ther Clin Risk Manag. 2009; 5: 427ï¿½448, "The benefits and risks of testosterone replacement therapy: a review"