Are there banks accepting bitcoins? We can open Dollar and Euro accounts in the banks. Can we open it on Bitcoin's account? Are there banks supporting this?
Banks would only do that to earn money from it, which means they'll be providing loans in Bitcoin, for which they could even start fractional banking again: lending out more than they have. Let's not!
One of the perks Bitcoin offers, is being able to control your own money by yourself. Nobody can take it from you, nobody can freeze your funds, and only you have the private key.
I can't think of a single thing a bank can do to improve Bitcoin for me. Even if the bank is insured for theft, you'd still need to secure your account and computer. While you're at it, you could just as well secure your own Bitcoins.