True fractional reserve banking requires ... loaning out the same money twice or more.
Sure you can do that with bitcoins. Imagine if MtGox went into the lending business. They lend money to someone, and suddenly that person's MtGox balance shows some bitcoins, without MtGox needing to accept any more Bitcoin deposits.
If everyone who borrowed money from MtGox withdrew their money, that would cause a run. But if borrowers just used it as a "bank account" for paying to other MtGox accountholders, there would be no run.
With any kind of fractional reserve banking, if there is a run someone loses out. If MtGox went into the fractional reserve loans business and suffered a run, the losses would be borne by MtGox or by its creditors or by its depositors. When a bank suffers a run, the losses are borne by the bank's shareholders, or by its creditors or depositors, or by the taxpayer.
I think i see few problems with running a fractional reserve bank with bitcoin:
1. In the internet era information flows very quickly, so the probability of a bank run is high if people start to suspect something.
1a. It's too easy to call the bluff with bitcoin. If people want to take their funds out of the bank, they can do it instantenously. Much easier than with gold. So it is bank run is even more likely.
2. To run a fractional reserve bank in any country of the world, you need to cooperate with local authorities or you will be closed down. And since authorities will not understand what Bitcoin is, there may be problems.
2a. The central bank would have to have reserves of Bitcoin for fractional - Bitcoin based system to work.... Nah, too complicated, not likely to happen in the closest future.
