While there's no rule against having as many pokémon with you as you want, in official league sanctioned matches, you can only use six. Most people don't want to keep pokémon they can't devote their attention to with them, so they make arrangements for the ones they're not actively training.
"By default, if you have six and you catch another, your dex will send your newest catch to a designated location. You can turn that up, turn it off, or even turn it down. I wouldn't recommend that last one, though. There's four options in Unova for where to send your pokémon. You can send them to your home, you can send them to my lab, you can send them to Amanita's PC, or you can send them to Professor Sumac's manor. I'm afraid that if you don't have a permanent place of residence in Unova you really only have three choices. If you send them here, I can take care of six pokemon, completely free of charge, and still take the time to give them some attention every day. It's not many, but they'll be in good hands.
"If you send them to the PC, you can store thirty for free, and pay a minor fee for each additional thirty spots, so to speak, but they'll just be living in their ball at a secure location, rather than really being taken care of. As for Professor Sumac, she runs a facility in the City of Northeast, where she has space for as many pokémon as you care to send her, however, given that she allows all of the pokémon on her estate to socialize freely, she does reserve the right to refuse service to any particular pokémon at any time. Please let me know if you have any questions."
