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.
No doubt it will happen but be sure to bend according to the policy set by bank to any Bitcoiners that would be depositing Bitcoin into bank, a lot of strict regulations will be passed to complete KYC verification, minimum and maximum amount and more which will actually control the customers hindering their full potential.
Also, the withdrawal and deposit fee banks can be charging will be too high.
Banks always make perfect calculations. There is no change in the value of fiat money in their own country. So the bank can deduct the payment fee from its customer. But the fee of Bitcoin or other cryptos goes up and down depending on the network speed. So it will be difficult for the bank to maintain these issues. And since it takes time for the Bitcoin block to be confirmed, the bank will not be able to make instant payments to the customer. But it is not impossible, when people understand the issues, then according to the bank's policy, Bitcoin deposit and withdrawal can be started. But in this case, Bitcoin will be centralized, those who will withdraw Bitcoin deposits directly from the bank will have no privacy.