Why can Pokemon learn only four moves at a time?

They don’t, anonymous. Not really. It’s just that sometimes, learning new tricks builds on concepts used for old tricks, so it’s easier both for the trainer and the pokémon to stick to newer techniques (or a certain set, at least) and build on those. The reason why is because … have you ever learned something advanced and then stumbled or had to think for a moment about something in the same field that’s far simpler? For example, have you ever gotten so used to solving complex algebraic equations or calculating tips on the go that when faced with a third-grader’s word problem, you had to stop and think for a second to recall old concepts? Pokémon do the same thing when trying to recall moves they haven’t used in a while, and those few seconds can be crucial to a match. On the other hand, attempting to reinforce new moves as well as old holds pokémon back and prevents them from mastering the new properly, so it’s either be a jack of all trades (and master of none) or rely on newer moves and avoid that moment of confusion when attempting to use old.

So in other words, it’s just easier for trainers and pokémon to recall a set of four specific techniques than it is to recall every technique they’ve ever learned. (Why four? Oddly enough, this seems to be the ideal number for a vast majority of pokémon. No one’s quite sure why.)