If you look close at some "fake" jerseys, you will see they are the real ones, made in the same sweat shops. The difference is the NFL has a markup of about 700% on the jerseys.
That is a fallacy. I worked in the fashion accessory field for a few years, doing product design and development. I was in a bunch of these big Chinese factories a lot, some that made legit licensed product for major labels, and some that did the knockoff trade. It would not be as simple as you think to just run an extra few thousand jerseys. There is a lot of accounting going on. I know a guy who manufactures the trim for official jerseys for all the leagues. Those orders are placed and controlled by reebok/nike etc and sent to the asian factories. They know what goes in as far as material and what comes out in finished product.
Some fakes are better than others, but there is very little to no chance a fake is secretly real. this goes for licensed team apparel and fashion items like handbags, and even ugg boots and North Face jackets.
And for those wondering what the actual difference is in terms of quality with fakes, it is generally in color matching different fabrics and in the trim pieces. The trim pieces especially, as the official guys go to great lengths to not have the shrinkage of the trim mismatch the shrinkage of the rest of the garment. If you have ever seen a jersey with arm bands looking tight and causing the sleeve to blouse, that is an issue with the trim shrinking to much in the wash. That shouldn't really happen with the officials.
Looking at the funky materials nike is making their jerseys out of, it will be a little while before decent copies using materials like that pop up.