You can only carry six for now, you'll have to hand deliver any Pokemon beyond 7 back to a "Pokemon HQ" or "Pokemon OutPost". You have to deliver the Pokemon before 24 hours or the Pokeball's battery will die. 

EDIT: And we'll say that, if a Pokeball's battery would die, the Pokeball would open up and release the Pokemon back into the wild. It's not like the Pokemon dies inside the ball. lol

This is something they would have went over when they gave you the Pokeballs. You can write as if your character heard this news already.