But this is impossible under modern monetary theory!

Or perhaps we should revise that to a "this is impossible under a deeply deluded understanding of modern monetary theory". For there's a certain segment of the populace who insist that banks just make up money out of thin air. So, therefore, this can never happen:

Ordinary Greeks rushed to withdraw cash from ATMs in the early hours of Saturday morning. Greece's Alpha Bank stopped all online transactions according to its website on Friday night.

If banks do just create money ab nihilo then this cannot possibly happen. There is no possibility of a bank ever running out of money, is there? But this is happening. Therefore it cannot be true that banks do indeed just create money out of nothing.

The confusion comes from the way in which credit is created: this is indeed done by the banking system in a fractional reserve banking system. You or I go to borrow money and the money we borrow is indeed simply created, as a ledger transaction, by that bank at that time. So, to some that seems the end of the matter. But at 4 or 4.30 that afternoon, that bank has to balance its books. It must have sufficient deposits to fund all of its loans, and if it does not through its branches it must go out into the more general market and solicit some more deposits. So, that effortless creation of money only lasts until that daily point at which it must balance the books.

And, of course, the same occurs in reverse when people are reducing their deposits at said bank. It must either claw back the loans it has made (something that takes time) or it must collect more deposits from the wholesale system or it must deny people the right to extract their deposits. Because, once a day at least, those books must balance.

In a world where banks effortlessly print or make as much money as they wish banks runs cannot happen. We are seeing a bank run: therefore banks cannot effortlessly print or make all the money they wish. Monetary theory's just great but even that has to be checked against reality occasionally.