As most people already mentioned, a bank in the sense of a trusted authority that controls your coins seems a step backwards. However, holding a savings account with a bitcoin "bank" could be a nice thing to have: I would earn interest X% and the bank would lend at X+Y% to cover lending costs and make a profit.

I'm not sure how exactly the ownership of these savings could work. Perhaps the coins would be outright sent to the bank and some token distributed in return (which could create a separate market) or a 1 of 2 key setup.