No team had ever won four postseason games prior to the 1980 NFL season, although was just the third year of the wild-card round, so the opportunities to do so had been pretty limited. It was the 1980 Oakland Raiders that were the first team to do what is now fairly common, as they took the long path to a Super Bowl win.
The 1988 Los Angeles Rams put their fans on a roller-coaster ride. They began the season in that made it look like they might join a parade of champions in their home city. The Lakers won the NBA title. The Dodgers were en route to winning the World Series. USC and UCLA each stayed in serious contention for the national championship. But the Rams then went through a stretch where they looked ready to miss the postseason altogether, before ending up somewhere in the middle.