Yeah, if you're using a concession ticket make sure you have the required proof (according to the rules on transport.vic.gov.au) for the specific type of concession you're claiming, and that you have it with you, or you'll get a fine.
Also "Student" is only for school students, full-time tertiary student eligibility is for a "concession".
When they're doing barrier exit checks (mostly city stations) they'll be checking the concessions only because the barriers make a different sound/light for a concession myki.
Otherwise the only extra hassle is having to show your relevant concession ID with your myki when inspectors are doing the rounds; I always liked to take forever to find a valid ID ("Oh, is that one expired? Wait I think it's in here...") to delay them and give fare evaders a chance to escape.
I think the half-price fares are worth the hassle, and health care card in particular is useful for other stuff, e.g. some GPs will bulk-bill then.