Can I invest tax saver FD in a bank and mutual funds simultaneously?

You can invest in Tax Saver FDs which have a lockin of 5 years. But the returns are much less than an ELSS or Tax Saving Mutual Fund. The differences are:

1. ELSS has 3 year lock in vs Tax Saver FDs having 5 years
2. Returns are 15% long term in ELSS where as FDs have half of that. as a result an investor having a long term horizon will get 4x to 5x more corpus by investing in ELSS mutual funds.

Make sure you invest only on Direct Plan Mutual Funds, else the broker makes a commission off your investment leaving you poorer by 40% in the long run!