Even if BTC replaces the usd or the euro or the yen , people will still flock to deposit their btc in a banallet (bank-wallet) owned by a third party.
Why would they? You definitely can, but unless you have total control over your wallet, it's not your money. I don't see much need for a bitcoin bank at the moment, online wallet (where you still have the wallet, service only stores the blockchain) - sure. Exchanges, definitely. Banks, as in services where you send your money to store them - have not seen any surviving long enough to prove useful yet.