Most of us can sit back and not worry about our cash because most of it is insured by the FDIC (Federal Deposit Insurance Corporation), created 75 years ago during the Great depression.
The FDIC insures your cash accounts, money markets and CDs for up to $100,000 total, per bank. If you have a joint account, it is insured up to $100,000 for each person in the account, that is in addition to your individual accounts.
The FDIC says if your bank goes under, it first tries to find another bank to take it over. If that happens, all of your accounts simply transfer to the new bank and you have full, immediate access to your money.
If no other bank is interested, the FDIC has two options. Either it creates a new bank operated by the FDIC, as it did when IndyMac in went bust earlier this year (you usually can get to your money by the first business day after the bank closes) or the FDIC simply cuts a check to each depositor for the insured amount, which usually happens within three days after the bank closes. So either way, you have access to your cash very quickly.
The FDIC notes that in 75 years it has handled the failure of more than 2,200 financial institutions, and nobody has lost even a penny of their insured deposits. So this is about as safe and sure as it gets, but only if your money is in FDIC-insured accounts and you do not go over the limit.