Of course it's real money. The entire global economy is built on this.
They can't run out of money if they create it ex nihilo. They can only run out of borrowers
It is interesting to discuss what is REAL money
Many people thought that their bitcoins on MTGOX are real, since they could sell them for fiat and use fiat to buy them on MTGOX. But as long as they don't withdraw, those bitcoins are just numbers on MTGOX's database
In fact those coins might have disappeared as early as 2011, and MTGOX were just playing a game of mixed FRB and using new customers deposit to cover the old customer's withdraw
This is very similar to what banks are doing. Bank's settlement network can be regarded as a giant exchange similar to MTGOX, where every people put their fiat in and trading against each other (in fact their goods are not listed on this exchange, banks purely exchange numbers between different accounts)
When you withdraw fiat money on an ATM, you get real money (cash notes and coins). In a system of 100% backing, numbers on your bank account is backed by real money, but in reality, banks all run FRB. And when they are running out of money, they go to FED asking for new money to pay the old customer
Banks will never run out of borrowers, they borrow from each other, and that is their daily business. But since the aggregate real money is limited, no matter what kind of borrowing activity they carry out, they will run out of real money as a whole in a liquidity crisis (liquidity means real money's liquidity), without real money, they go bankrupt
I modify the dinofelis's word a bit here: Bitcoin is "real bitcoin" if enough people think of it as "real bitcoin". You will clearly see this is not true, bictoin is real bitcoin if you can see them on blockchain
Unfortunately, in today's fiat money system, there is no tools like blockchain to let people inspect the flow of real money, they can only believe what banks tell them, only exchange owners (e.g. banks) know how much real money they have.
That is the reason when the banks all had a liquidity problem, they will create a large shock in economy, just like MTGOX did. Unlike MTGOX, their scale is millions of times larger, if they failed, every one on this giant exchange will lose shirt, thus they must be bailed out by FED using real money, and FED's real money as discussed in RBD theory above are in fact created out of nothing, to make the whole scheme even more unreal