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 BowlsXVII, 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 1983, Washington won the eastern division of the NFC with a 14-2 won-lost record, thanks to the same offensive line, leading the NFL with 541 points scored (33.8 points/game). 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 Oakland Raiders, losing Super Bowl XVIII. Although they won the eastern division of the NFC again in 1984, they lost the divisional round of the 1984-85 NFL playoffs to the Chicago Bears. The Redskins became a force again in 1986 with a 12-4 record, winning a wild card game against the Los Angeles Rams and the divisional round of the 1986-87 NFL playoffs against the Chicago Bears with 138 and 134 rushing yards, respectively, but could neither pass (150 yards) nor run (40 yards) much against the New York Giants powerful defense, losing the NFC title game.