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.
Definitely yes, there is really a high possibility that this will happen though no one can tell when in particular it would happen, can happen in 5, 10, 15 or 20 years from now, or may even happen sooner than any of us think, it all depends on individual countries and what their policy on bitcoin would be.
But most important to know and note is that banks are not important for bitcoin, depositing bitcoin into banks is like we running back to the same problem bitcoin was created to solve after embracing bitcoin, it makes no sense, banks had and still have custody of our fiat money, they control it, if we should now start depositing bitcoin to the same banks we should be running away from because of their monopoly, what makes us think they will not start controlling bitcoin one day as well?
Best way to enjoy bitcoin is to allow it remain decentralized as it is, hold and manage your own coins by yourself, you will always be more at peace with yourself when you know that you will always be able to make use of your bitcoin when exactly you need to without waiting for an approval from any body or entity.