A Bitcoin bank? I think that won't be possible. Even tho if any bank would do that, I'm pretty sure there will only a few people that'll interested by it.
In my opinion we already have a Bitcoin Bank!
Coinbase is doing exactly the same things a bank.You store your coins there and use of them for gambling, they will shut down your account immediately.I think they also had idea to loan out customers coins if I'm not wrong.