Sure. They can be turned down. (BTW: Also remember USMC and NOAA. )
As reflected by Dodberb, different requirements for different military branches.
Although there are probably differences in medical and performance characteristics, here's one example ... (I don't have specs, so just go with example, and not the specifics--other Medical / experienced folks can jump in here)
To get into USMC, you have to run 3.0 miles is less than xx:00 minutes.
Other branches have a 1.5 mile requirement.
Our Mid, who meets qualifications in all other areas for all service branches, takes off on his qualifying run -- the last hurdle, so to speak.
At the midway point (1.5 miles), he has a qualifying score ( less than xx:00/2 ) for branches other than USMC. He finishes his run, but logs in at XX:45
45 seconds too long for USMC).
So, our hypothetical Mid would be INelegible for the Marines, but would be able to qualify for the other service branches.
In short, he'll still need to meet the requirement of the branches of service he wants to enter. I think this is the same for any other Academy , or ROTC graduate -- you need to be qualified to enter the branch upon GRADUATION as well.
There are real-world examples (none at USMMA that i'm aware of), of hopefuls riding a motorcycle, celebrating ... and having an severe leg injury which completely eliminates ability to run and qualify for any military branch. This is also why ALL the Academies don't want their kids riding motorcycles, or recreational skydiving (afa DOES do skydiving).
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In the meantime, however, you mid is going to be more concerned on his upcoming test.... Get the grades, study.. GET to graduation!