In the NFL, which team has had the greatest time in possession but still lost the match?
The '96 Rams, or the '09 Dolphins
According to Statmuse.com, the 1996 St. Louis Rams lost this game to the Baltimore Ravens, despite having the ball for 45:43. If you're limiting the question to non-overtime games, then the record belongs to the 2009 Miami Dolphins, who lost this game (at home, no less!) to the Indianapolis Colts despite having the ball for 45:07.
(The Statmuse database goes back to the 1991 season so there is a possibility games before that may have greater time in possession)