I don't know how they did it, but I was checking around for fun starting with 1999 in the draft history section from a couple of sites, and saw that there were four teams with at least two first round picks.
Then in 2000, there were three teams, including the Jets with three picks.
In 2001, there were two teams, with the Rams having 3 picks.
In 2002, there were two teams.
In 2003, there were four teams.
In 2004, there were five teams, including the Texans.
In 2005, there were four teams.
In 2006, there were three teams.
In 2007, there were two.
In 2008, there were an astounding number of five teams.
In 2009, there were four.
In 2010, there were another four.
In 2011, there were the Saints.
In 2012, there were four.
And in 2013, there were three, with the Vikings amassing 3 picks.
There were several with multiple first round picks in consecutive seasons, too.