This is simply not true. Usually bank deposits are insured by the central bank or another such agency, but there's a limit (e.g. $100k per account).
> The convention and social contract has been that you deposit money for safe-keeping in a bank.
There's indeed a social contract of sorts, but since the most of the world converted to fractional-reserve banking, banks are not pure deposit banks anymore.
In fact there have been many bank runs around the world, even in the US.
http://en.wikipedia.org/wiki/List_of_bank_runs
It's well-known that individuals who own more cash than the insured limit should spread it over several bank accounts. Of course, that is moot when the government simply confiscates your money.