The Buffalo Bills have missed the playoffs for 15 years now. In the era of free agency, that is unprecedented. The Bills are even approaching the record since the AFL-NFL merger, when the league implemented the Wild Card round of the postseason.
After the merger in 1970, each conference had three divisions - but the league wanted an even number of playoff teams in each conference so they added what would later be called the "Wild Card" team as the best of the non-division winners. Prior to that, only division winners had made the playoffs in both leagues (and the playoff droughts are much longer due to the scarcity of spots).
The New Orleans Saints began their existence with 20 consecutive non-winning seasons. Only 17 of those seasons were in the Wild Card era, marking the longest drought in the time span from 1970 to 1986 before finally cracking the barrier to get to 12-3. Until 1978, only one Wild Card team made it in each conference, but expanding the field wouldn't have helped the hapless Saints.
From 1983 to 1997, the Arizona Cardinals went 15 years without making the playoffs. After their 1998 Wild Card appearance, the team went nine more years without a playoff berth. Without that lone season, they would hold the dubious mark.
In 2014, Buffalo tied the Cardinals at 15 seasons. To make matters worse, the Bills haven't really been close to the playoffs all that much in the streak. They haven't cracked the top seven spots in the conference in that span, and were only mathematically alive entering the final week of the season once (2004).