This is possible when you can create commodities out of thin air (like "paper silver").
I have thought of making a proof-of-concept "Casascius Bank" but where I accept no more than about 1000 BTC in total deposits.
The real goal, though, wouldn't be to become a bank. It would be to illustrate a proof-of-concept: provably secure depositing.
I had a similar thought (create proof-of-concept bitcoin bank). I would also have auditable deposits. However I would give out ripple BTC IOUs and have a transparent reserve ratio. 100% in the beginning, but less later.