Maybe I'm not thinking this all the way through yet, and haven't wrapped my mind around something I'm missing yet... but I got to thinking, how many prohibited persons are there?
So I did some searching and found The FBI records total page
And saw that more than half were illegal aliens, #2 was adjudicated mentally deficient at just over a third of their records, and limping WAY behind at #3 was the the criminal category. Now, I thought to myself, maybe there are multiple reasons, its easy to imagine there are felons who have been adjudicated mentally deficient. But when you add every category that isn't an illegal alien (You'll see why in a second) that's still less than half, or barely more than 4 million people. The reason we don't count the illegal aliens is because they can't vote. And over here at this research website Pro-Con (As in pros and cons of an issue, not go-convicts-yay!)
we find Nearly 6 million people are disenfrachised for a felony conviction. This is staggering to me. Even if every single person in every single non-alien category were a disenfrancised person, that still leaves a missing million and a half or better people not in the NICS database.