Quite frankly Dave, fatigue is often used as a convenient excuse for crews that fail to apply common sense, and actually watch what the hell they are doing.
I have seen the young inexpereinced as well as older guys fail to notice what the numbers are for various weights, and apply what you have called a gross error check.
This is especially apparent with FMS/FMC equipped aircraft...crew members look at the numbers, and accept them at face value, even though they couldn't possibly be right....forgeting the garbage in/garbage out principle.
Does fatigue play a factor?
Yes, sometimes...but I believe more often it is simply carelessness.
Example.
A well rested crew, beginning a long range flight, planned at max weight.
The Flight Engineer hands up the takeoff data card, which shows the correct loadsheet weight alright, but the V speeds selected are woefully short of whats required, and likewise with the thrust selected.
I handed the card back to the Flight Engineer and mentioned...you don't really believe this, do you?
He looks at it for about three seconds and says...'ah, well....No, opps, sorry Captain, wrong page.'
Precisely why the data should be closly checked by at least two folks.
FD crew should always be prepared to ask...does this look REASONABLE, and if not, do something about it, pronto.