The Washington Nationals / Montreal Expos franchise has the longest current streak of seasons not making the playoffs. They have not made the playoffs since 1981.
The Boston Celtics
Deleted previous one. Lakers did not make the playoffs in the 2004-2005 season. The real answer is the San Antonio Spurs, who have made the playoffs every year since 1998.
Right now it's the Mariners. Until this year, the longest streak was held by the Royals and the second-longest the Blue Jays...but the Royals are in the Series this year and it is what it is.
New England Patriots---The patriots lost their last game in the playoffs
Longest streak goes to BYU Do not currently know who has the longest current streak