It should work just fine if they do the investments in Bitcoins.
Below is an entertaining, albeit slightly alarmist, educational video on how the money system works today. This should be required viewing for anyone considering using money:
In particular, banks do not lend out deposits, at least not in any meaningful way, and the so-called money multiplier is non-existent.
To get up to speed on such things, I would recommend the blogs of economics professor Bill Mitchell. His style is a bit rambling without much polish, since he writes one entry every day, but I found it very worthwhile for getting a deeper understanding. Consider looking for entries with "money multiplier" in the title in the index here: http://bilbo.economicoutlook.net/blog/?page_id=1667
What most people don't realize is that banks don't have to actually have the money they lend out. That's the basis for our monetary system and while it has nice qualities, like spurring innovation, there are some undeniable drawbacks.
(Obviously I'm simplifying, your money gets pooled with others and deposits and withdrawls happen from a pool of money, you don't get your own money back when you withdraw, but lets compare two hypothetical banks with only 1 customer each)
A dollar denominated bank takes your money and invests it. They use it for mortgages, or they put it in bond markets or equity markets. They make a profit. Some of that profit gets returned to you as interest. They also provide for safe keeping of the money. With interest rates near zero, we forget this, but thats the real reason for a bank. If it was just a safe place to keep money, we'd be paying them.
A Bitcoin bank couldn't work. You give them some Bitcoins. They transfer them to dollars, make some investments. Now you go to withdraw some money and they have to sell the investments. Since Bitcoins are deflationary, they can not make you whole without incurring a loss. There is no money to be made in operating a Bitcoin bank and giving interest so long as the exchange rate of Bitcoins to the real world keeps going the way its going. Everyone should just sit on their bitcoins and hold them.
Now if you just want to make a safe deposit box for Bitcoins, that's would work. And you would have to pay a fee. But there is a difference between a bank account and a safe deposit box, and I don't see bank accounts working with bitcoin any more so than they would have worked based on tulip bulbs in the 80s.