You can’t check the torque of a tight fastener by trying to turn it with a torque wrench. Well, you can, but if it doesn’t move, it doesn’t tell you much. It might be under-torqued by 20% but not move, because of sticking friction.
And that’s fasteners that you’ve just set. Ones done yesterday or last year or last decade might be farther out by corrosion or lack of grease.
For a torque wrench to give a good result, the fastener has to be moving, free of stick-slip, smoothly and slowly at about 20–30 degrees per second, with a long final swing to target torque – ideally a quarter of a turn. That’s 3 or 4 seconds for that last quarter turn. Slow.
A lot of people swing the wrench too quickly, bang into the end of the free travel after the click, ignore the problem when a bolt jerks or creaks, don’t give a long final swing (sometimes you have to back it out to redo the final swing from a better starting position once you know roughly where it will end), and don’t exercise a click-type wrench half a dozen times at the target torque or higher if they haven’t already used the wrench that day.
These things matter more for accuracy than whether the torque wrench is specced to 2% or 4% accuracy in a lab test – where all of these details, and others, get the attention they deserve.
I had farming neighbours as a child in the UK, which might by why I tend to over-torque things when left to guesswork. Massey Fergusons from half a century ago could take a lot of abuse.
But I have noticed, as dnschmidt says, that most people under-torque instead. This is especially pronounced with people who have desk jobs and little mechanical experience. Many of them barely snug up the fastener, leaving it at comically low torque and therefore tension. That ruins things just as surely as excessive torque; it just takes a little longer before the damage becomes apparent. And it might be even more dangerous, because something falling off in operation can be worse than a bolt breaking in the garage.