Is the military allowed to discriminate?

All the branches have minimum and maximum height limits.

Yes, but your phrasing allows for no wisecrack.

Anecdote warning.

While working for a Federal agency (law enforcement), and BSing with the recruiter, he told me a story about an applicant who had applied and been rejected for having been convicted of a DWI. That demonstrates alcohol abuse, and can mess up one’s background investigation.

Long story short, the dude sued claiming that, as an alcoholic, he had a disease that qualified under ADA.

Agency’s response: Unfit for duty. Go f*ck yourself.

Agency wins.

One thing you have to understand about the ADA is that it ONLY requires an employer to make a REASONABLE accommodation. It doesn’t mean you have to hire that person if you needed to accommodate him/her in such a way that would mean a major “unreasonable” accommodation.

What is unreasonable? Well a judge decides that. This is why an employer will never tell you why you didn’t get a job.