I've done some messing around and I think what's happening is that if you can't compete for some reason (for example, you have the wounded trait), the game doesn't tell you that you can't compete. So you try and get qualifications, and the game lets you, but since you aren't allowed to compete they get set to zero right before the start of the competition
I've found that a good way to tell if you are allowed to compete is if the bar says "You did not qualify" before the competition starts, you have a trait that stops you from competing. If it says "You are not likely to qualify" then you are allowed to compete, but you need more qualifications. I would appreciate a system to actually inform me of what I can and can't compete in, ideally before travelling to a tournament.