- Bank runs. It's standard practice that banks will loan out more money than they have in deposits, because it's unlikely that all their loans will go bad at once, and/or it's unlikely that all their deposits will want their money back at once (this is known as fractional reserve banking[0]). But if your currency is anchored to something fixed, this can happen. FDIC insurance is guaranteed in the US because even in some absurd financial economic meltdown scenario, Ben Bernanke can print out USDs to make every depositor whole, if necessary. The only "cost" to that kind of "bailout" is the potential inflation. But you can't do this as a government if your monetary supply is limited because it's fixed to something you can't directly control. The government can borrow money and then make depositors whole, but those that you borrow money from may impose conditions, and those conditions may exacerbate the problem you're trying to solve. See: Cyprus[1].
- The deflation-debt spiral[2]. Yes, inflation "punishes savings," but deflation punishes debt. Generally, debt becomes a big problem in recessions. If you owe $20,000 on your car, and you get a 20% wage cut because your company can only sell goods at 20% of the price they were previously able to, your debt does not go down to $16,000. It's still $20,000. So more of your income will go to paying off debt, which means less of it will go to discretionary spending, which means even more economic slowdown, which means more deflation, which means more wage cuts, which means debt becomes an even bigger burden, etc.
The only real danger to a fiat currency is whether your government's central bank will be so irresponsible with the money supply that inflation spirals out of control. If you're a resident of a third-world country like Zimbabwe[3], transacting in bitcoins will be great because you'll happily trade potential deflationary pain for not being at the whim of a dictator that will print money to increase inflation to one thousand trillion bazillion percent. But if you're the resident of a first-world country which has a responsible central bank, then increasing the monetary supply to fight off deflation is a very good thing.
[0] http://en.wikipedia.org/wiki/Fractional_reserve_banking
[1] http://www.theatlantic.com/business/archive/2013/03/why-the-...