Wouldn't really make sense for a bank to accept Bitcoin unless they see a reason that they'd be losing out on a lot of potential revenue due to not accepting it. Like, lets just say that a lot of people wanted to pay their interest on a loan for some sort of item in Bitcoin I highly doubt they'd be caring (if the customer paid the fee both ways) and they'd be doing fine with this sort of system.
Though, a bank wouldn't be accepting Bitcoin as a way to deposit / withdraw as it wouldn't have any sort of gain for them in the least.
I think it take a long process, we all know that all the bank transactions supervised by the Government, bitcoin is anonymous so anyone who trades with bitcoin surely must get government approval.
This too, as the banks are going to be highly regulated by the Government who has a vested interest in not allowing Bitcoin to thrive. The governments are going to be the people who ensure that Bitcoin isn't accepted and furthered by people.
So no, no real national / regulated bank is accepting Bitcoin nor will they probably in the near future.