I'm sure there was a question like this before, but I can't find it.
In the Pokemon games, the game doesn't allow you to carry more than 6 Pokemon at any one moment as you travel around. You can own more than 6, but the ones you aren't currently keeping in your party and actively using have to be stored in the PC. (How are living beings stored in a PC?... That's a whole other question...)
Most explanations I have seen have said that the Pokemon league only lets you in if you only use 6 Pokemon, but not all trainers are aiming for the league or even anything since they just train Pokemon for self-defense or companionship. In this case, why would they need to stick to 6 if they aren't aiming for the league?
(Similar limits exist in other games. For example, in lots of RPGs, you can only take 4 people with you on a quest even if you have recruited more than 4. The remaining people have to be benched or stay in an inn. Answers could address more general cases than just the Pokemon games if they wanted to, but should mainly focus on this particular Pokemon case.)