I dunno about Navy - but when I was at Air Force basic, they had the "Pararescue" (PJ) tryout day, I think it was around week 4 or so. There was one guy in my flight who joined up strictly to be a PJ. It was all he talked about. It was all he wanted to do. I don't remember what all he had to do - run, swim, situps, pushups, etc. All I remember is that for one of the tests (I think MAYBE it was swim), he just barely missed whatever the standard was by a very small margin. Didn't make it into PJs. When the list of random assignments came out, he ended up being assigned as a groundskeeper. Long way to spend a 4 year enlistment.
We had another guy who wanted to be a F-15 mechanic - he ended up being a groundskeeper too. Unlike the PJ-wannabe though, this guy was a complete moron and frankly, I think keeping the grass cut may have been more than he could handle. Even so - rough way to spend his 4 year enlistment.
Hint: the smart ones get their job assigned and guaranteed (as much as you can trust the government anyway) BEFORE they enlist. Doubt you can do that with the specialties like PJ, Navy Diver, etc. though. If you really want to go in, you might want to get a good job guaranteed, and then try out for the divers. That way, if you don't make it, you're not completely boned.
Good luck!
AKDiver