I dont buy thé idea of bitcoin bank for now.
We should be thinking about the life growth of bitcoin not the bank.
Bitcoin bank can not help bitcoin for now and it will make the popularity of bitcoin to fed away.
Actually already banks started implementing online services completely and try to compete with the bitcoin. Citi bank, New York creating new digital coin for them called "Citicoin". However, none implementation are equal to bitcoins. If bitcoin adopted by any financial services. They will consume more amount for usage and fees. Personally I do not support this bitcoin in bank.
I have noticed that this thread already contains a really big number of different answers from particular people, but I just had to participate at least for a while.
It is happening because banks are into technology and current financial news, and they are scared of cryptocurrency horizon that is starting to cover some of their clients.
They can spot an incoming struggle with bitcoin, and that is why they are implementing so many new online services.
For example, Ripple has been created by banks to make the settlements between them.
Also, as you said there is Citicoin, but I cannot believe it will be adopted by a big amount of users, when we have so many great cryptocurrencies to choose.
Im sure that banks will take part in cryptocurrency business, but in few years because I think they are still not sure about going into investment that is not so predictable.