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Just a handful. You rarely seem to come across opponents with a full team of 6, so, if they do have a lot of Pokemon, they're not being used in battles.
Also, despite the number of trainer battles you have in a game, you don't see many people with many Pokemon in their homes -- maybe something small like an Espurr or Fletchling, but they're more like a family pet than a travelling companion at that point. So, again, if people do collect a lot of Pokemon, they'd be kept primarily in PC storage, it appears.
