I could live with 14 teams in the playoffs. I looked on
pro-football-reference.com at the standings since 2002 (the year the NFL went to 4 divisions in each conference) to see what the records of the 7th playoff team would have been if it was a 14 team field each of those years. In that time, 4 teams would have gotten in with 8-8 records versus 6 teams that were 10-6 or better. All the rest would have been 9-7.
Adding a 7th team in each conference would actually mitigate the problem created by 4 divisions where you occasionally have a very weak division sending a team to the playoffs.
Here's the list - I didn't try to figure out tie-breakers since I was focusing on the record as opposed to the specific teams. I also included the teams that would have had to play in the 1st round since the #2 seed wouldn't have had a bye.
2002: Dolphins, Pats, or Broncos (all 9-7) and Saints (9-7). #2 seeds: Titans and Buccaneers.
2003: Dolphins (10-6) and Vikings (9-7). #2 seeds: Rams and Chiefs.
2004: Ravens or Jaguars (both 9-7) and Saints (8-8). #2 seeds: Pats and Falcons.
2005: Chiefs (10-6) and either Cowboys or Vikings (both 9-7). #2 seeds: Broncos and Bears
2006: Broncos (9-7) and either Packers, Panthers or Rams (all 8-8). #2 seeds: Ravens and Saints
2007: Browns (10-6) and either Eagles, Vikings or Cardinals (all 8-8). #2 seeds: Colts and Packers
2008: Pats (11-5) and either Cowboys, Bears or Buccaneers (all 9-7). #2 seeds: Steelers and Panthers
2009: Texans or Steelers (9-7) and Falcons (9-7). #2 seeds: Chargers and Vikings.
2010: Chargers (9-7) and Buccaneers or Giants (10-6). #2 seeds: Steelers and Bears.
2011: Titans (9-7) and either Eagles, Cowboys or Cardinals (all 8-8). #2 seeds: Ravens and 49ers.