This is going to be a short answer, and it's no. The historian Alan Williams conducted tests with various weapons and armor as part of his book The Knight and the Blast Furnace. I don't believe he tested for knives, but he found that a sword was incapable of generating enough force to pierce mail. I imagine the same would be true for a knife, only more so. Mail is really very good armor. The only weapons Williams found to be capable of reliably defeating mail were powerful arrows, lances, and polearms. You can probably add maces and crossbows to that list.

How, then, to defeat an armored man? While our literary descriptions of high medieval combat are very sparse, the artistic evidence is interesting. If you look at the Winchester and Maciejowski Bibles, two illustrated manuscripts from this period, you see a couple of basic strategies. One is to strike your opponent where the armor isn't - on an exposed shin or wrist. Another is to grapple and get under/around the armor with the knife. But either way, you're not hacking or stabbing your way through steel rings and the padded undergarment beneath them with a pocket knife.