So just figured I'd pipe up and offer some help, given that this site has been so helpful for me for years (going back to college and determining where to go to law school). This thread of course has been useful, so in case my experiences can be of use, feel free to send a pm/throw stuff out there.
I graduated law school in 2011, and unfortunately was not picked up on any of the student programs. I applied to the Navy 5 times, AF 3 or 4, and Army twice; only time I wasn't flat out rejected was as an Army alternate (which, no offense to Army folks, but thankfully I wasn't picked up on).
I re-applied for both the AF and Navy last year, and was picked up for the Navy in the Spring, and recently have commissioned, heading to ODS in a few weeks. I did a Navy internship during my 1L summer, and they were far and away my first choice, so it ended up working-out.
To throw some random thoughts out there:
-Obviously, if at first you don't succeed, try try again. But I would also say to analyze why you aren't perhaps getting in, and if there's anything you can do to control it. Personally, I re-worked my motivational statement several times, as well as sought out opportunities (a dual-degree in criminal law, and a post-law school job as a federal prosecutor with 12+ trials) that were applicable.
-Medically/background wise, be honest. I had some stuff that wasn't entirely helpful, and may have hurt/delayed my process, but in the end it's better off being revealed now than later.
Anyways, thanks for the help to the random JAG folks on here, and good luck to those who are trying to get there.