One day banks will accept Bitcoin as medium of exchange and accept deposits from bitcoiners. As soon as the world powers accept it as medium of exchange.
This could happen one day when banks can no longer compete with Bitcoin because everyone uses it like they currently use fiat money. But as long as fiat money remains trusted by all governments around the world, it will continue to be used by everyone, even if everyone loves Bitcoin as much as they love money. Banks could take advantage of this opportunity to earn fees when people buy or exchange Bitcoin through their services. However, I still prefer to stick with certain exchanges if I only want to exchange Bitcoin for other assets like fiat money.