[2] Bostic was one of "The Hogs", the Redskins' famed offensive line which also included guards Russ Grimm and Mark May, tackles Joe Jacoby and George Starke, and a few other new additions over the years.

The team defeated in succession in the 1982–83 NFL playoffs the Detroit Lions, the Minnesota Vikings, the Dallas Cowboys, and the Miami Dolphins, rushing respectively for 175, 204, 137, and 276 yards, with mainly John Riggins as the running back.

They beat the Los Angeles Rams in the divisional round of the 1983–84 NFL playoffs and the San Francisco 49ers in the NFC championship game, passing for more yards than the previous year, but also with some more hefty running: 130 and 172 rushing yards, respectively, but they could not push off so easily the defensive line of the Los Angeles Raiders, losing Super Bowl XVIII.

The 1987 NFL season was another strike year and another Super Bowl win for the Redskins, with 15 games played instead of 16 and owners sending in replacement players.

With an offensive line composed of Bostic, Raleigh McKenzie (replacing the injured Russ Grimm again) and Mark Schlereth at guards, Jim Lachey and Joe Jacoby at tackles, the Redskins beat the Philadelphia Eagles in a wild card game, with 93 rushing yards, but could not get by Joe Montana and the San Francisco 49ers in the divisional round, gaining only 80 rushing yards, as the Redskins had to pass to try to catch up to the 49ers' potent offense.

The following year, Bostic, due to an injury, played in only 4 of 16 regular season games and was replaced by Raleigh McKenzie in the 1992–93 NFL playoffs, the Redskins beating the Minnesota Vikings with 196 rushing yards, but losing, as they did in 1990, to the San Francisco 49ers.

On October 25, 2015, Bostic was inducted into the Washington Redskins Ring of Fame at half time of the game versus Tampa Bay, which the team would win 31–30.