It doesn't seem to make sense for people to store bitcoins with a third party.  It goes against the entire premise of bitcoin, which is peer-to-peer transactions without intermediaries like banks.  It makes much more sense to manage your own keys, according to the degree of security/paranoia you personally feel comfortable with, which probably varies a lot depending on how many bitcoins you actually have.

Bitcoin online banks (online wallets) are not regulated, may not have deposit insurance, and do not pay interest -- so they're fairly different from fiat banks, and present different kinds of risks for someone depositing their keys there. 

For the masses, I can see that people won't want to manage their own private keys -- it seems too techie for them.  That's not true, but the masses tend to go for the easiest solution.  I can see the masses, if they ever really do adopt bitcoin to a larger degree, gravitating towards online wallets due to the ease of use and similarity with existing online fiat banking.  But for people who are enthusiasts -- which pretty much describes most people holding bitcoin currently -- it doesn't seem to make sense to hold your coins in a bitcoin bank -- goes against the entire philosophy of the system.