I highly doubt there would be a bank for bitcoins because the whole point of btc is for it to be anonymous and that bank will totally destroy that ideal.
I agree, i think it wont happen because it will be a big problem for their dominance. Bitcoin is anonymous and yes but that doesnt mean it will not be good to be know by other because as far as I observed here many users are hoping that bitcoin will be legalized if thats happen then it wont be anonymous anymore. Bitcoin is not accepted by bank but you can cash it out there.