Right. That is something than can be built on top of bitcoin, bringing traditional banking services to it. If the problem is managing your private keys, that can be delegated to a bank that you can trust - which will recover them for you in case you lose them, paying a fee, for example. That necessarily means that the bank has access to your wallet though.

Bitcoin even offers a bit of extra support for that too, via transparency, based on the public blockchain. There is embedded protection against third parties: if a rogue employee of the bitcoin-bank transfers bitcoins out of your bank-wallet, the transaction can be followed. It might not be possible to recover the money, but at least some kind of identification can be performed - with the help of the bank and with legislative support, this can be enforced. This transparency is not there for tradicional banks: if there is a problem, you trust them to solve it, but you can not independently see what happened to your money.

This relies on the fact that anonymity is a fragile concept in the bitcoin world: an address is anonymous as long as nobody puts the effort (and has the right extra pieces of information) to correlate address to person.

At the end of the day, it will be a matter of the bank having a business interest in keeping a clean reputation. This is too early for the bitcoin economy, so I would not trust any online bitcoin bank with more than a couple of bitcoins. CitiBitcoinBank is not here yet.