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I never said it wasn't possible. But since there are so many pokemon I wanted to use, I decided to use more than 6. Plus, I can keep them more diverse and I can use more types of moves and combinations.
It's called rotation. You train more than 6 pokemon and rotate the extras in and out whenever you need them whether it be for a type advantage, HM use, personal preference, training, etc. Some of the more advanced trainers use that method. According to Zeb, it's pretty easy to use more than 6 in BW2.