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.
Money invested in banks in India is very safe. This is because the government regulates and monitors the functioning of all banks to ensure that banks do not cheat their depositors and that they do not engage in high risk activities that can cause such heavy losses that they will not be in position to repay their depositors. Further, there is a provision for the assets and operations to be taken over by the government in case of any credible risk of bank failure.
In spite of these precaution, some scares of failure of a smaller bank may be created once in a while. However in the last 50 years there has not been a single incident of a bank failure or a depositor loosing the money put in the bank.
Any bank operating in the US has to be a part of the FDIC, or the Federal Deposit Insurance Corporation. This is a federal body that provides insurance to depositors. Whenever a bank fails and is unable to pay back to its customers the money they have deposited in the bank, the FDIC secures deposits up to $250,000.
So your money deposited in a bank is safe up to the limit of $250,000. Banks are required to pay a fee to the FDIC for the services it provides and follow some rules and regulations which have been set up to ensure that there is adequate liquidity and a bank has enough resources to repay its depositors.