Yes, they do. I strained my achilles tendon back in July of last year, and it continually got worse to where I was in a walking boot for three out of six months. I was able to waiver the October PRT, but this time I was asked to visit Navy Medical at FW NASJRB to get their evaluation on whether the injury would be a long-term issue or just a short-term. I was told that if the issue was deemed to be long-term and a potential risk, they could attrite me as NPQ and let me go. Of course, this is worst-case scenario, so they would likely try to rehab you first and work to get you better, but I've heard of several individuals who have had this situation occur to them.
I finally was able to get into physical therapy back in April (problems with Tricare and my PCM and ortho communicating delayed me three weeks) but I'm not 100% yet, so I will likely waiver the run next week barring anything out of the ordinary at JRB. I have been told that you must do a full PRT (run, PU and SU) before going to OCS, so I'm steadily working out and getting my run distances and times up before end of June/July gets here (I go 17 July). A month ago, I could've tried to run the PRT but extreme pain would kick in around 400m...now (after a month of therapy and the walking boot, etc) I'm able to run up to half a mile no problem but get tight, but it's the tightness of not having run in awhile and not from the injury. In another month I should be able to run the 1.5 miles - it won't likely be a dazzling time, but I will run it under Good-Low. I've never been really injured other than sore muscles or minor aches and pains, etc. so this has been a trial for me.
Long story short, don't get injured; if you do, stay on top of it every second and keep your OR and the chain of command informed at every chance. The more you pass on, the more informed they are of your situation and the less likely it will come as a surprise later on. I complete my collegiate contact reports bi-monthly and email my OR at least that much to keep him up to date, so as long as you don't injure yourself in a way to automatically DQ yourself, and you keep on the rehab and keep Navy informed you should be fine. Anything else that you can't control (like my Tricare-PCM issue) is out of your hands, so do everything you can do and it will work out.