Of course it is possible to operate a bank based on BitCoins. And it that sense, then there can also be more than 21 Million BitCoins. That is because the bank doesn't hold the money you give to the bank, it can lend that money to somebody else while you don't need it. So you might believe you "own" 100 BTC, because the bank owes them to you, but in reality somebody else has your 100BTC - so there are two people who believe they are worth 100BTC, but there are only 100BTC as a whole...
There will never be more than 21 Million Bit Coins, it's impossible. You can 'promise to pay the bearer' or loan but the amount in circulation will never go above 21 Million.
Real banks in most countries have a "brake" installed by law, afaik: they have to keep a minimum fraction of the money that people deposit. Otherwise they could in theory create an infinite amount of money, as long as people kept depositing everything in the bank.
Which is exactly why banks for bitcoin are a bad idea. When push comes to shove whoever owns the kitty owns the lot and it's tough luck to anyone who trusted them. Eventually they'll push the limit and won't be able to pay their members back anymore and boo-hoo. Since one can easily take a simple loan out outside of bitcoin, bitcoin loans will follow the worldwide loan market since we can exchange bitcoins for money and so on. Therefore, since bitcoin has no real advantage for loans and a great deal of risk, loaning will have little effect on the value of bitcoins. If bitcoins were designed to be loaned, they weren't designed very well.
