Like T.L. Jones said, they are not supposed to, but its a basic assumption that they do. If you look at when shootings happen they always seem to know when a shooter had a background check performed through NICS. I'm not sure if there are other methods in which they can track and detect that sort of thing, but I've always just assumed its because they have a good idea of who has what at what times.
And there is no trouble to buying multiple guns in a short period of time. You can by 1 or 50 guns all on the same background check if you wanted.