Sadly, the tolerances are as inhumanly small as they stated. The intuition is that the fractional change in orbital period will equal the fractional change in orbital speed (times 3, but we're only interested in orders of magnitude).

Thus, in low earth orbit, at a speed of 8000 m/s, a relative speed of 8 mm/s is a fractional change of 1 in 1 million. The orbital period is 90 minutes, a millionth of which is half a second. That doesn't seem like much, but objects in orbit move really, really fast. Arriving at the same spot half a second later means you'll miss the other object by 4 kilometers! Without active guidance, a rendezvous in orbit is near impossible.