Yes and no.
I wouldn't say they ever diagnose. My PTs have diagnosed me correctly, but it still required an MD for confirmation. They can't diagnose, but they can highly suggest that this is probably what the diagnosis is, and then send to an MD/DO for confirmation. PTs cannot (as far as I know) order MRIs, Xrays, or CTs, which a lot of the time is required to properly diagnose a patient. My PT could tell I had tendonitis in my FHL, but the MRI showed not only that, but also tenosynovitis of the FHL, plus 3 other tendons that hadn't yet showed symptoms. I wish I could see a PT without a prescription though, because my tendonitis acts up a lot and I hate having to call MDs to get a prescription. Fortunately now I've worked at a clinic long enough that I can in essence treat myself (do the exercises I did in PT, ultrasound myself on my lunch break, talk a PT into helping release tight muscles/tendons), and I know MDs well enough who will fax me a prescription if I need one since I've seen them millions of times. It's still annoying though. lol
In the case you described, the PT would probably send the patient to an MD anyway, just to make sure it's not something else disguising the problem. How much would it suck to treat a patient for back pain only to find out they really had 3 herniated discs that needed surgery, or a problem with their kidneys that the patient thought was muscular. These are extreme circumstances, but not totally impossible. (Although PT wouldn't do much good in this case, so it would get figured out sooner or later).
In some states PTs can see patients without having an MD referral/prescription ("direct access" states). It is my understanding that in these states people can go to PT like they would any other doctor. I'm not totally clear how that works with insurance - I'm guessing they still cover it, and that the PTs give a more generic diagnosis code (like back pain, tendonitis etc.), as opposed to the super specific ones a patient might get if they see an MD for a diagnosis. However I believe insurance companies can deny the claim if they don't think it's medically necessary, in which case an MD diagnosis can help support the necessity of the PT.