A 1980 graduate of Clemson University, Bostic was named All-ACC in 1979 as an offensive lineman for the Tigers. He was also named to Clemson's All-Centennial Team in 1996, and inducted into the school's Hall of Fame in 1997. In 1999, he was placed in the South Carolina Athletic Hall of Fame. Bostic has won more Super Bowls (3) than any other former Clemson Tiger.

Bostic was a member of the Washington Redskins from 1980 to 1993, helping them win Super Bowls XVII, XXII and XXVI and becoming the only Clemson player to play for three Super Bowl Championship teams. He was also named one of the 70 greatest players in Redskins history. 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.

In 1981, in his second year, he replaced Bob Kuziel as the starting center for the Redskins in all 16 games. He continued in that position until his final year, 1993, though missing important amounts of games because of injuries in 1984, 1985, 1987, and 1992.

In 1982, the strike-shortened season at 9 regular season games, the offensive line was potent throughout their 8-1 won-lost regular season, with Bostic at center, guards Russ Grimm and Mark May, and tackles Joe Jacoby and George Starke starting all 9 games. The same line prevailed in the playoffs, except that Fred Dean (offensive lineman) replaced May at right guard because of an injury in games 1,2, and 4. 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. The Redskins won Super Bowl XVII with Bostic and the interior line bowling over nose tackle Bob Baumhower and the two interior linebackers A. J. Duhe and Earnie Rhone.

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