I was in the Army infantry for 4 years, so I know a little about this.
Honestly it depends on your unit. Most of what the army teaches is BJJ based. They switched over about 10 years ago when they learned that BJJ was easy to learn and easy to apply, even in combat gear. (In a novice sense compared to intricate standup)
When you go into basic training if you're in a combat MOS like the infantry they'll put you through about a week's worth of "combatives" as they call it. Every drill sergeant places a different amount of importance on the combatives experience, so at some basic training companies you might not get as much exposure. I was lucky because 3 out of 3 of my drill sergeants were airborne guys from 101st, 82nd, and 173rd in Vicenza, Italy. Those units are generally thought of as more "high speed" and accept the combatives training a little more. Especially 82nd Airborne, which is at Fort Bragg, NC.
At Fort Bragg there is the actual combatives school (there's also one at Fort Benning.. which is where if you were in the infantry you'd go to basic/OSUT) the one at Ft. Benning is where the rangers train at. To be an actual full time army combatives guy i really don't know what you'd have to do. Or if you could do it. However, you can get certified to be a combatives instructor. It works on levels, and as you got higher and higher in levels I'm sure you could eventually become a full times combatives instructor (which would be sweeeeeeeeeeeet)
Personally I got stationed in Germany, and when I got there my unit was already deployed, so there wasn't that many people to train with. Although you could usually go to the gym and pick up a training partner to roll around the mats with. Once the unit got back all army combatives stuff went to the wayside and the stupidity began. I think maybe 2 or 3 times over the course of the next 2 years we did combatives training, and this is an infantry unit.
Finally we got deployed again, and while we always wrestled around amongst each other, there was never any actual "army combatives" training in Iraq. There was especially never any combatives with Iraqis.. cuz, well.. if they got that close to you they'd just cap your ass if you didn't get them first, haha.
During all that time, and after I got back from the deployment (another 6 months) I did a couple of Army boxing smokers, worked out a TON, and yet we still never really did combatives. I never once saw a germany wide combatives competition. We'd have them within the battalion (about 450-600 men) but that was few and far between. All in all for 4 years of the Army the only combatives training/experience guided by the Army I had in that whole time totalled up to maybe 3 weeks. Including basic training week long combatives. The rest of anything I accomplished during that whole time in a martial sense was completely on my own.
I know it was long winded but the whole point of the story is to show you that even if you think you're joining an organization that will utilize all of your martial abilities and help you develop them (even in the most front-lines combat unit you can get) you're probably wrong. There is too much risk associated with letting guys manipulate each other's joints/bones and unit commanders will put it to a stop quick. I'm not sure about Green Berets and Rangers, since I wasn't one, but I can only imagine it's very similar. Moreso than you would believe. Everything you do in the military will have a risk assessment associated with it, (even 3 day vacations) and if the numbers don't match up the unit commanders will put a stop to it. They'll run you 10 miles up hill every morning til your legs want to fall off and the entire unit has shin splints, but they can't set aside 1 hour to train armbars. The square wheel that keeps rolling.
By the way for my credentials I wasn't exactly a "shitbag" guy either. I was sniper trained, in the best squad in the battalion (won 3/3 squad awards on FTX's) won an Army iron man competition, and I have about 10 solid years of martial arts experience. So I wasn't the kind of guy they'd pass over for good training.