This is patently untrue. If you're driving in a 70 mph zone and you're going 71 mph, that's illegal. Of course, if you can make a judge doubt that the cop's radar gun was calibrated correctly it can't be proven beyond a reasonable doubt that you were speeding, but 71 in a 70 is still illegal.

That said, any cop who would pull someone over for a 1 mph infraction is a mega-douche.