Not from the anime, but the manga does suggest an answer in Volume 10 of the Pokemon Adventures series (in the Gold&Silver arc).
Professor Oak: Of course it's possible to carry seven or eight Pokemon - even twenty or thirty, if for some reason you must. But how could oen trainer give that many Pokemon enough individual love and attention? Only we researchers can justify big collections. Among trainers, anyone who uses over six is disparaged.
Professor Elm: Of course ...
Professor Oak: Which is why the Pokemon league decreed 6-to-1 as the ideal Pokemon-to-trainer ratio. It's the perfect balance for a serious trainer. And Crystal is clearly serious!