Question

What team has gone the longest without having a 1,000-yard rusher? For the team in question, the player with the last 1,000-yard season is still active in the league.

Answer

Carolina, with Jonathan Stewart running for 1,133 yards back in 2009.
Two other teams have gone over five years, the Browns (Peyton Hillis, 2010), and the Jaguars (Maurice Jones-Drew, 2011).