Yes, one NFL team has gone 0-16 in a regular season. That team is the 2008 Detroit Lions.

The 2008 Lions were a team in disarray. They had finished the previous season with a 7-9 record, and expectations were high that they would improve on that mark in the 2008 season. However, the team suffered a series of setbacks in the offseason. They lost several key players to free agency, and their head coach, Rod Marinelli, faced criticism for his handling of the team.
Things only got worse once the season began. In the first game of the year, the Lions lost to the Atlanta Falcons by a score of 34-21. They followed that up with losses to the Green Bay Packers, San Francisco 49ers, and Chicago Bears. By Week 5, the Lions were 0-4 and things were looking grim.
Despite the team’s struggles, there were some bright spots on the roster. Quarterback Jon Kitna had put up decent numbers in the first few games of the season, throwing for over 800 yards and five touchdowns. Wide receiver Calvin Johnson was also having a good year, with 293 receiving yards and four touchdowns through the first four games.
However, the Lions were unable to translate individual performances into team success. They continued to struggle on both sides of the ball, and by midseason, it was clear that they were in danger of becoming the first team in NFL history to go winless in a 16-game season.
The team’s nadir came in Week 11, when they lost to the Carolina Panthers by a score of 31-22. The game was notable for a few reasons. First, it was the closest the Lions had come to winning a game all season. Second, it featured an infamous moment in which Kitna, who had suffered a back injury earlier in the game, attempted to return to action. He was promptly sacked on his first play back, and had to leave the game for good.
The Lions went on to lose their remaining games, finishing the season with an 0-16 record. They became the first team in NFL history to go winless in a 16-game season, and their futility made them the subject of national ridicule.
In the aftermath of the 2008 season, the Lions cleaned house. Marinelli was fired, and the team brought in a new coaching staff led by Jim Schwartz. They also made several key acquisitions in the offseason, including drafting quarterback Matthew Stafford with the first overall pick in the 2009 NFL Draft.
The changes paid off in a big way. In the 2009 season, the Lions won their first two games of the year, snapping their 19-game losing streak. They finished the year with a 2-14 record, but it was clear that the team was moving in the right direction.
Since then, the Lions have had their ups and downs. They have made the playoffs a few times, but have yet to win a postseason game since the 1991 season. However, they have also avoided the kind of catastrophic collapse that led to their winless 2008 campaign. They will look to continue their progress in the years to come, and hopefully, avoid the kind of ignominy that they experienced in 2008.