Why does DES use exactly 16 rounds?
From Schneier's description of DES in Chapter 12 of Applied Cryptography (12.3): 


  DES with any number of rounds fewer than 16 could be broken with a known-plaintext attack more efficiently than by a brute-force attack.


This explains the "Why not less than 16".

As for the "why not more than 16", that is a tradeoff for speed of execution (more rounds = less speed).