The NFL.
The Dallas Cowboys but this was back in the 1970's so idk now
The Indianapolis Colts hold the record for the last decade, 9 appearances in the last 10 years. 2000, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009. They did not make the playoffs in 2001. If they make the playoffs in 2010, they will still have 9 appearances in the last 10 years, and will tie the record of the most consecutive playoff appearances in the NFL.
Bills 4
Between 1993-2010, that is the Cincinnati Bengals with 2 playoff appearances (2005 and 2009). The Cleveland Browns have also had 2 playoff appearances but there was no team in Cleveland from 1996-1998. The Houston Texans have no playoff appearances but they entered the NFL in 2002.
The New York Yankees with 47.
the Montreal Weiner schnitzels
Florida with 34
Texas tech
the new york knickerbockers, with 13
UCLA with 13!
UCLA with 10, Cincinnati and Duke with 5