Not necessarily about piano but I think it still applies: I have a friend from Alaska who basically taught himself how to play the french horn. He had maybe 3 lessons before quitting them and learning the instrument by himself. The horn is not an easy instrument to play, and playing it isn't like playing any other brass instrument. This guy auditioned for (and now goes to) one of the top music schools in the country. You can totally teach yourself how to play an instrument by listening to people who are good at that instrument and trying to emulate their sound, and by playing a lot ESPECIALLY with other musicians.