Bolden attended Scotlandville Magnet High School in Baton Rouge, Louisiana. As a senior, Bolden rushed for 886 yards and 12 touchdowns en route to being named First Team All-District 5-5A. He also caught 12 passes for 256 yards. SuperPrep named him SuperPrep All-Region and ranked him the No. 34 best player in Louisiana. Bolden was named the No. 93 best running back in the nation by Scout.com. He was heavily recruited by Les Miles and LSU, but ultimately decided to play for Houston Nutt at Ole Miss.
Although fellow running back Enrique Davis was much more regarded as a recruit, by his sophomore year, Bolden had gained the starting role for the Rebels. In the 2010 season, Bolden averaged nearly six yards a carry. Against BYU in the first game of the 2011 season, Bolden suffered a broken ankle. Although the fracture was a minor one and he only missed a few games, he never returned to his previous form. He was not selected in the 2012 NFL Draft, although some sites predicted he might be drafted in one of the later rounds. Although Bleacher Report reported on April 30, 2012, that he had agreed to a free agent contract with the Washington Redskins, Bolden later tweeted that the report was false. On May 2, 2012, a New York Times blog reported that he had signed a free agent deal with the New England Patriots.
During the 2012 preseason, Bolden had 36 carries for 128 yards and caught four passes, but fumbled the ball twice, losing one. Nevertheless, at the end of training camp, the New England Patriots elected to keep Bolden on their 53-man roster, rather than try to place him on their practice squad. He scored his first touchdown in the Patriots' Week 3 loss to the Baltimore Ravens; in Week 4, he led the Patriots with 137 yards on 16 carries—an average of 8.6 yards per carry. He scored one touchdown and added an 11-yard reception. He and Stevan Ridley, who had 106 yards on 22 carries with two touchdowns, became the first Patriots running backs to accrue 100 rushing yards in the same game since 1982. Moreover, Wes Welker and Rob Gronkowski had 100-yard games that day as well, making the four only the second such quartet of two 100-yard rushers and two 100-yard receivers in a single game for the same team. For his efforts, Bolden was named the NFL's FedEx Ground Player of the Week for Week 4. On November 9, 2012, Bolden was suspended four games for violating the NFL's policy on performance-enhancing substances. Bolden was active for 12 of the Patriots' games during the 2013 season and finished the year with 271 rushing yards, 3 rushing touchdowns and 4.9 yards per carry; he also had 152 receiving yards.
Bolden was a part of the Patriots Super Bowl XLIX championship team when they defeated the Seattle Seahawks, 28–24, and played in all 19 of the Patriots regular-season and postseason games. On January 9, 2015, Bolden was signed to a two-year extension by the Patriots. On November 9, 2015, Bolden caught his first touchdown pass, an 18-yarder from Tom Brady, against the Washington Redskins. On November 29, Bolden caught his second touchdown pass on a 63-yard catch-and-run in a loss to the Denver Broncos. In the 2016 season, Bolden went back to a reserve role on offense and played mostly on special teams. On February 5, 2017, Bolden was part of the Patriots team that won Super Bowl LI. In the game, he played only on special teams as the Patriots defeated the Atlanta Falcons by a score of 34–28 in overtime. The Patriots trailed 28–3 in the third quarter, but rallied to win the game, 34–28, in the first overtime game in Super Bowl history and the largest comeback in Super Bowl history.
On April 5, 2017, Bolden re-signed with the Patriots. On September 2, 2017, he was released by the Patriots, but was re-signed two days later.
Brandon is married to his high school sweetheart, Arianna King. They gave birth to their first child, Brycen, on May 2, 2011. Brandon's grandfather, Frank Pitts, was a wide receiver who played for the Kansas City Chiefs in Super Bowl I and Super Bowl IV.