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.
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The Miami Hurricanes had a total of 6 NFL players drafted in the first round of the 2004 Draft. They were
5) Sean Taylor - S
6) Kellen Winslow Jr. - TE
12) Jonathan Vilma - LB
17) D. J. Williams - LB
19) Vernon Carey - OT
21) Vince Wilfork - DT
The Hurricanes also had 4 in 2003, 5 in 2002 and 4 in 2001. In a 4 year span, THE U had 19 first round draft picks.
Not sure if it transfers into real life, but in madden 2011 (Which is has been very accurate on the rules of the NFL) the maximum number of draft pick one team can have is 9.