Wes Welker went undrafted in 2004 out of Texas Tech. He made the Chargers roster as a rookie tryout but was released before the first game. The Dolphins signed him in 2004 where he played for 3 years before being traded for a second rounder to New England.
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No ... he went undrafted in the 2004 NFL draft and signed a contract with the San Diego Chargers prior to preseason that year.