Putting on and using MOPP gear (anti-NBC suits and gas masks), at least gas masks from what I remember, is something that's taught in boot camp and you do get periodic refreshers afterward, but it's hardly something that a lot of time is spent on. The thing that you don't see on TV in movies is all of the misc other training that goes in the military that doesn't directly relate to your job, esp. nowadays, Before major holidays you'll have to attend classes/briefings on drinking and driving. Then there's the mandatory sexual harassment classes and with the current socio-political climate, there's probably classes on racial and cultural sensitivity as well. On top of that, lower enlisted will often get pulled for things like cleaning duties, helping out in the mess hall, playing gopher, and a whole host of things you'd never think you;'d be doing when signed on the dotted line.
TLDR, the military has a million and one things they expect their people to do that don't necessarily apply directly to combat, even for those in combat arms. So troops having difficulty with NBC gear is not entirely unrealistic. Neither is getting a refresher once in theater on a combat deployment when an NBC attack is a possibility.