ok i run a quick check : played 215k (6max) hand this year, been dealt KK , 922 times, lost to AA 9times, won 1. and i fold KK twice preflop.
first i will only consider a fold if opponent as a low PFR <5% (6max) and i would never fold it to stack below 60BB regardless of stat.
here is the 2 hands i folded:

NL 200(6max)
5 handed

UTG ($601.25)
Hero ($384.60)
Button ($32.50)
SB ($280.40)
BB ($29.00)

Preflop: Hero is MP with , .
UTG raises to $7, Hero raises to $24, 3 folds, UTG raises to $70, Hero folds
Reason : villain stats (27%/5% 2.5) Deep Stack, small 4 betting.


Hand 2 :
NL 200 (6max)
5 handed

CO ($148.70)
Hero ($194.50)
SB ($389.80)
BB ($211.80)
UTG ($197.00)

Preflop: Hero is Button with , .
UTG raises to $10,CO folds, Hero raises to $27,SB folds, BB folds, UTG raises to $187 and is all-in, Hero folds.
Reason : raise and push from a very tight, very passive villain ( 18/1.5/1.2) from UTG.

in none of these cases i get to see their cards so, i dont really know if i was right....