Reaction time is irrelevant here, because you're anticipating a response to your input, not triggering the stopwatch in response to unexpected events. Human sensitivity to timing is much better. Great drummers can hit the beat to within a few milliseconds. If you've watched the movie "Whiplash", think of the "rushing or dragging" scene.
Following a beat will give you very consistent times, but they can still be off! Try with a stop watch and try to stop it at exactly 10 seconds (you are free to look at the clock), you'll probably be 10-30ms off. Then try stopping it at an arbitrary time, but the same every time, and you will probably stop it at the exactly same time every time!
Suppose the human is allowed to reject trials where they feel they botched the timing?
Let's say the clock flashes an LED every second, and you are trying to press a mechanical button that will make an audible click on the 10th LED flash.
To decide whether or not to accept a trial, the human has to judge whether or not the audible click and the flash of the LED were simultaneous. I have no idea how good we are at that, but I'd expect that we are better at that than at reacting to things.
Even better would be to make the button also flash an LED, so that there is no error from differences between aural and visual processing speed and latency.
See http://journals.plos.org/plosone/article?id=10.1371/journal....