According to the Pro Football Hall of Fame, two teams with an 11-5 record have missed the NFL playoffs during the 16-game regular season era:
The Denver Broncos (pictured) posted an 11-5 record in 1985 which placed them one game behind the Los Angeles Raiders in the AFC West. Denver lost out on the playoffs via tie breakers to the New York Jets and New England Patriots who also posted 11-5 records that season. [...]
The Patriots suffered a similar fate in 2008 when they finished 11-5 and missed the playoffs while the 8-8 San Diego Chargers won the AFC West. Although New England had the identical record as Miami and Baltimore, they lost out on the division crown to the Dolphins and the second wild-card spot to the Ravens based on conference records.
Prior to the 16-game regular season era, there were three teams that finished with 11 wins that failed to make the playoffs, according to this ProFootballTalk article:
Three 11-win teams missed the playoffs in the old days, before the 16-game season and the expanded playoffs. The 1967 Baltimore Colts missed the playoffs at 11-1-2, the 1963 Green Bay Packers missed the playoffs at 11-2-1 and the 1962 Detroit Lions missed the playoffs at 11-3.
As for the worst team to make the playoffs, this Bloomberg article discussing Week 17 of the 2010 season says that only three teams with losing records have ever made the playoffs:
The Seattle Seahawks are going to the National Football League playoffs with a losing record, the first time that’s happened in a full season. [...] The Seahawks won the National Football Conference West Division by beating the St. Louis Rams 16-6 last night at Qwest Field in Seattle to finish 7-9 for the season. [...]
The only [other] time a team with a less-than .500 record made the NFL playoffs was 1982, when a players’ strike shortened the season to nine games. The Cleveland Browns and the Detroit Lions both reached the postseason with 4-5 records.
A 7-9 record has a .4375 winning percentage, and a 4-5 record has a 0.444 winning percentage, so that means the 2010 Seattle Seahawks at 7-9 have the worst record to ever make the playoffs.