NFL teams are greedy.
As they should be.
Since there is no such thing as having too much depth, and since the proverbial diamonds in the rough are found year after year, front offices don't even pause their talent search upon the draft's completion. As soon as that door closes, it's on to the next one, as executives and scouts comb through the ranks of undrafted free agents in hopes of finding a keeper.
It happens more often than you might think. Tony Romo, Warren Moon and Kurt Warner weren't drafted. Same goes for Antonio Gates, Wes Welker, John Randle and James Harrison.