Is there some reason for there to be more groups with 16 conjugacy classes than with 15 or 17?
It is a well-known exercise to show that a group with one conjugacy class has only one element, a finite group with two conjugacy classes must be cyclic of order two, and a finite group with three conjugacy classes must be cyclic of order three or non-abelian of order six. At some point I (along with everyone and their brother) classified finite groups with four conjugacy classes, but I never really looked beyond that until today.
My results are only partial, but I couldn't help noticing local maximums in my census at 10 classes (gentle), 14 classes (medium), 16 classes (sharp), and 18 classes (medium), with corresponding dips at 11 (gentle), 15 (sharp), and 17 (sharp).
Off hand I can't think of why a group might be more likely to have an even number of classes than an odd number, but perhaps this is well known. Four and five classes are relatively well known, as in one of my favorite papers, Miller (1919).