Bradley, who became an influential figure after replacing the injured Ray Allen in Boston's starting lineup, had been hampered by his left shoulder all season.
"It's disappointing. He's been terrific this year," Celtics coach Doc Rivers said on the team's Twitter feed.
Bradley, 21, has averaged 7.6 points per game for Boston this season while making 41 percent of his three-point attempts. He has been especially impressive on defense.
However, he missed the last two games of Boston's Eastern Conference semi-final series against the Philadelphia 76ers after his shoulder kept popping in and out.
The Celtics and 76ers are tied at 3-3 in the best-of-seven series with the decisive game to be played Saturday in Boston.