No i do not think because bitcoin is made out of the fact that people do not trust banks why should we put something like this in a bank i think that is the dumbest question i have ever seen you cannot ask something that is made to never put in a bank to put int in a bank.
Banks suck and they will always suck. The community will not even accept it and there is even no way they can easily operate in the first place. What would be the idea? To help you store your money in their bitcoin wallets and give it to you when you want which you can easily do all by yourself. Or probably to give you loan as the OP mentioned, which is total crap.
Having a bitcoin bank would be so useless and no one will even bother to come up with such an idea in the first place. It is even hard to trust centralized exchanges, let alone a bitcoin bank.