They found that 8mm was enough to stop a heavy machine gun firing AP. The real problems only began when bazookas started showing up in numbers.
8mm was generally sufficient against standard small arms and light MGs, but dedicated small-caliber anti-tank rifles with tungsten penetrators were still able to punch through at not far short of normal engagement ranges. Either the tank had to sit well back and support its accompanying infantry from a distance, or else it was at serious risk of having the ventilation system supplemented by a few extra holes.
10mm and up quickly became the norm for frontal armor in several countries, but side armor was more likely to be struck at angles, increasing its effective thickness, and so remained at 8mm on a lot of AC and light tank designs. The increase in frontal protection rendered portable AT rifles ineffective in most situations, and most armies gradually phased them out.
20mm AA guns (and similar AT designs) were generally more than sufficient against pre-war light tank designs, and still more-or-less viable against a lot of pre-war mediums. As Germany discovered to its dismay, not everyone had kept the thinner armor, and German gun performance against some of the heavier French (Char B1bis) and British tanks (Churchill, Matilda) proved to be a problem.
Bazookas and Panzerfausts didn't show up until mid-war or later. Even by the end of WWI, everyone was well aware that you needed airfields to send planes up, and AA guns to bring them down.