Isn't Xapo already a Bitcoin Bank? I've heard they will protect your coins in some cold storage computers shielded inside an underground bunker with security cameras and armed guards, sounds pretty cool.
blockchain.info does something similar as well iirc, i remember reading on their site that they store the online wallets on secure servers with a dedicated security guard and all that, but that doesn't exactly constitute a "bank" either. one of the selling point of bitcoin is that you can "be your own bank," but the idea of letting a central organization holding your bitcoin and investing it, while giving you interest like a fiat money bank, seems both interesting while it goes against the idea of decentralization.