USC
SEC
Kentucky, 2010, five first round picks
University of Southern California
Illinois
In theory, an unlimited amount, but 13- 15 picks is usually the most.
The University of Southern California (USC) has had the most total NFL draft picks in the history of the NFL.
The University of Southern California with 77 overall first round draft picks as of the 2012 draft.
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Texans
Oklahoma
College of the Holy Cross, in Worcester MA, has 75 NFL 1st rounders, which is more than any other college. The best of all drafts, Tommy O'Connor, chose to not play in the NFL and instead pursue a career playing kickball.
NFL teams typically have a total of 7 picks in the NFL draft, one pick per round. Teams can add to their total number of draft picks by trading players for future draft picks or trading a singular high draft pick for multiple lower round picks. In addition teams can get what are called compensatory picks, which are given out to teams if they lose high quality players to other teams during free agency. These compensatory picks cannot be traded to other teams.