So, you can only train a skill with a trainer 5 times per level, but what if you miss a level or a few levels? Are you screwed out of training that skill for that level or does it add up?

No they do not accrue. If you only trained 3 times in the previous level, and when you level up, your training limit will still be 5 and not 7.

Is there a console command to fix this? Sounds like a pretty bad design decision.

Well you could do player.incskill to level up a skill by 1, just like you do while training. Refer to http://elderscrolls.wikia.com/wiki/Console_Commands_(Skyrim):Skill_List for the skill names.