It certainly is possible for a run on the bank to drive a bank into insolvency, exactly as you describe. However, there are some protections in place to guard against this.
There are regulations that specify a Reserve Requirement. Basically, the bank has to keep enough money around to cover nominal withdrawl demands.
If a bank does run into trouble, they can take out a short-term loan. Either from another bank, or from the central bank (e.g. the US Federal Reserve)
If all else fails and the bank can't meet its obligations (e.g. the loan fell through), the bank has an insurance policy to make sure the account holders get paid. In the US, this is what the FDIC is for.
These protections have worked pretty well for many decades. However, during the recent financial crisis, all three of these protections were under heavy strain. So, one of the things banking regulators did was to put the major banks through stress tests to make sure they could handle several bad financial events without collapsing. These tests showed that some banks didn't have enough money in reserve. (Not long after, banks started to increase fees and credit card rates to raise this additional capital.)
Keep in mind that if banks were unable to use the deposited money (loan it out, invest it, etc), the current financial landscape would change considerably.
- Free checking accounts would go away. Why would a bank want to assume the risk of watching your money if they couldn't get something out of it?
- Interest bearing deposits like Savings Accounts and CDs would go away. The bank is able to pay you interest because they are doing stuff with that money which yelds a profit for them (of which you get a cut in the form of interest).