Currently no banks accept Bitcoins in its form. They trade/transact in government backed fiat currencies only.
Although Bitcoin is a legal payment tender in some countries. Your traditional banks do not directly deal with Bitcoin. There might be some intermediary who convert Bitcoin to local currency and vice versa.
Secondly Bitcoins is against the concept of centralised authority in charge of your money. If you need a bank to park your money, you do not need Bitcoin.