Are nickels dated 1960 to 1965 worth anything?

Unless they're in uncirculated or proof condition they're generally only worth face value. Nickels with those dates still show up in change occasionally. In particular, huge numbers of 1964 nickels were struck during the great coin shortage of 1964-65, so they're the most common date from that period.

NONE of them contain silver; unlike dimes, quarters, and half dollars of that time, nickels were struck in the same alloy as today - 25% nickel and 75% copper.