Is the money invested in a bank deposit actually safe?

If you are in the United States, and if your bank deposit
is not too big ($250,000 and down), yes, it is completely
safe.

In the United States, the FDIC (Federal Deposit
Insurance Corporation) insures all bank deposits in member banks.  If your bank goes
broke, the government will pay you your money back.  This program was started in the
time of the Great Depression.  It was started so that people would not have to worry
about the security of their bank deposits.

For more details
about exactly what kinds of deposits are insured and for exactly how much, please follow
the link.