Usually, when people say they got more than 2 or 3 infusions, they aren't brewing it western style with just a teaspoon or tablespoon of leaves. In my experience, western brewing exhausts the leave in one steeeping.
Most of the time, anyone who says they got 3-20 infusions out the same leaves are brewing with gongfu brewing. Gongfu brewing uses a lot of leaves (1/5 - 3/4 of the vessel) and a small teapot or gaiwan, and the first steeping is usually around 5-30s (depending on the tea, and personal preference) and gets longer with each following infusion.
Another way to get 3-4 infusions out of the same leaves is "glass brewing," but that doesn't seem very common on this forum. I prefer this kind of brewing for green tea, and I usually get 3 infusions out of the same leaves. Basically, you put some leaves in a gaiwan or cup, wait a bit for it to brew, then drink it untill you have about 1/3 tea left. Then fill it back up with hot water, and repeat untill the flavor runs out.
If you want to get more than one infusion out of the same leaves using western brewing, try using twice as many leaves as normal, and cut the brewing time in half. Add a minute or two to the following infusions.