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.
Why would you want to send your Bitcoin to a bank when Bitcoin was literally created to work without banks and third parties? 
Even if they start accepting Bitcoin they would not allow you to withdraw whenever you want, they would charge additional fees, they would force you to use their ''approved'' wallet, and other crap like that.
Sending bitcoin to the bank defeats the purpose for which it was created. Bitcoin signifies "freedom of the unbanked," so what is the point of sending it to the bank then? Besides, the bank is a third party, and Bitcoin came as a rescue from third parties, and this also includes the banks which uses every opportunity it has to exploit people from their hard earned resources because of the fact that the government backs them up for their dirty acts. Even if the banks accept bitcoin, it will be the last thing I will ever do to deposit my bitcoin with the bank when I can keep it under my own custody.