First is the rate of interest offered by Treasury Bills. If the Treasury Bill rate goes down, as it has been in the past few months, bank fixed deposit rates too go down. Second is the risk associated with the bank. If the bank is perceived to be risky, then it would have to offer a higher risk premium to depositors, even though the Treasury Bill rate remains unchanged. The same applies to fixed deposits of companies.
Different companies offer different rates of interest on their fixed deposits.
When you invest in a fixed deposit, you have, in effect, become a creditor of the company. Each company has its own credit worthiness, i.e., the ability of the company to repay your short-term loan. Hence, you need to be sure that the company you are advancing money to is offering the correct risk premium. This could entail a detailed study of the company's annual report and other financial statements. Thankfully, there are independent agencies that do this cumbersome job for you.