Putney played college basketball for UMass Minutemen playing 131 games and starting 86 overall, with averages of 7.8 points, 4.8 rebounds, 1.2 assists, 0.7 steals and 1.2 blocks. His best year was as a junior, with averages of 10.1 points, 5.8 rebounds and 1.3 assists.
After going undrafted in the 2014 NBA draft, Putney was acquired by the Rio Grande Valley Vipers of the NBA Development League on November 2, 2014 after a successful tryout. On November 14, he made his professional debut in a 112–101 loss to the Idaho Stampede, recording two points, three rebounds, one assist and three blocks in 17 minutes. He was deactivated twice by the Vipers during the 2014–15 season, and subsequently played in just 24 games, averaging 6.3 points and 2.5 rebounds per game.
On June 13, 2015, he signed with the Perth Redbacks for the rest of the 2015 State Basketball League season. He struggled with his shot early on for the Redbacks, making just 20-of-62 from the field over his first four games. Despite his shooting woes, he was still effective on the defensive end, recording an eight-block game in just his third outing, and recording a double-double in seven of his nine games for the Redbacks. Over those nine games, he averaged 19.3 points, 10.9 rebounds, 2.0 assists and 2.6 blocks per game. He and teammate Ray Turner were a dominating big man pair but could not lead the Redbacks to the playoffs, finishing the season outside the top eight with a 10–16 win/loss record. Later that year, he joined Malaysian club NS Matrix and helped them win the 2015 Seri Mutiara Champions Cup. He also earned MVP honors for the championship game.
On November 2, 2015, Putney returned to the Rio Grande Vipers for the 2015–16 season and had a marked improvement in his second campaign for the team. On January 12, 2016, he was named NBA D-League Performer of the Week for games played Monday, January 4 through Sunday, January 10. Putney led the Vipers to a 2–1 week while averaging 28.3 points on 47 percent shooting to go with a league-leading 15.7 rebounds and 3.7 blocks. On January 29, he was named in the West All-Star team for the 2016 NBA D-League All-Star Game. On March 31, he was suspended for five games without pay for violating the terms of the league's Anti-Drug Program. The suspension ended his season with the Vipers, and in 48 games for the team, he averaged 17.1 points, 8.1 rebounds, 1.9 assists, 1.1 steals and 2.3 blocks per game.
On April 16, 2016, Putney signed with Guaiqueríes de Margarita of Venezuela for the rest of the 2015–16 LPB season. That night, he made his debut for the club in a 91–61 quarter-final Game 2 loss to Bucaneros de La Guaira, recording nine points, five rebounds and one block in 23 minutes. Guaiqueríes went on to lose the series 4–3. In six games for the club, Putney averaged 15.0 points and 4.8 rebounds per game.
On July 14, 2017, Putney played two summer league games for the Minnesota Timberwolves and Phoenix Suns. At 2pm PT, he went 0-5 shooting and grabbed a rebound in 4 minutes of game time for the Timberwolves against Washington Wizards and three hours later, after he was acquired by the Suns, where he scored 7 points, 7 rebounds and 2 blocks in 18 minutes of game time against the Sacramento Kings. On November 10, he signed with the Erie BayHawks of the NBA G League.
Putney signed with the Miami Heat on October 11, 2018. He was waived on October 13. Putney was subsequently added to the roster of the Sioux Falls Skyforce.
