Just posting what started as an amusing thought and not 100% in the know on the technical details of the bitcoin software as of yet to understand if this would be feasible but here goes....
Based on current bitcoin technology....
1) Public wallet system, stored in a P2P network
2) Block chains performing transaction validity
a) correct balance to transfer
b) valid destination and origination wallets
3) User creates account, private client issued to user.
a) Private wallet shows account balance
b) Provides authentication mechanism to initiate transactions
4) Miners block chains help ensure the security and functionality of the bank
a) Miners are rewarded with a nominal transaction fee for banking operations
Obviously a lot more could go into this, but this could help further the bitcoin cause if it is feasible.
Sounds like you are describing the existing Bitcoin network.