Is this a risk with Bitcoin? According to my limited understanding, it doesn't seem to be. That should in theory reduce the need for things like FDIC and complicated risk calculations, like counter-party risk.
MtGox doesn't seem like a "bank failure" or "bank run" to me. It seems like a bank that forgot to lock the doors at night.
PS. I'm not characterizing inflation as bad, nor am I a goldbug. I am merely pointing out that the entire population of FDIC-insured banks are paying constantly for the mystical "it might happen some day" promise of a bailout, due to an inflationary system that reserves priveleged access to a central clique - ie. it's not like the conservative viewpoint of "faith in <the national central bank>!" comes without strings. (I am most aware of this perspective as someone who used to live with someone on the interest rate committee for the Bank of England.)
Which may be a good reason to limit currency holdings to what you need to use as currency (that is, short-term medium of exchange for things of value), and invest the rest in productive assets, whether those are stocks, bonds, real estate, or digital collectibles like internet funbucks. But if you aren't directly holding those productive assets yourself but are trusting them to account held by someone else who promises to transfer them to you on demand or dispose of them at your direction, you probably want to do some consideration of their ability to keep them securely and execute the trust you are placing in them.
The British empire was built on the back of the Pound -- the first truly international, modern fiat currency. It reigned until it was muscled out by the fiat Dollar.
An immense amount of wealth has been generated for humankind on the backs of fiat currency.
And one more thing...
Bitcoin is also a fiat currency in the sense that it does not contain intrinsic value. It is only as valuable as its network effect. If the network falls in love with a different coin, bitcoin dies.