Brinkley played college football at the University of Missouri. In his four years at Missouri, he finished with 12 tackles. In his senior season, he recorded 5 tackles for the season. In his junior year, Brinkley was the team's long snapper for the third consecutive season. In his sophomore year, he was selected as the MU Special Teams MVP.[1] In his freshman year, Brinkley was selected as the MU Walk-on MVP.[1]
On April 30, 2012, Brinkley signed with the Tennessee Titans as an undrafted free agent.[2] He made his NFL debut against the New England Patriots on September 9, 2012.
On March 8, 2015, Brinkley signed a five-year contract extension with the Titans worth $5.75 million.[3]
On September 30, 2018, in a 26-23 overtime victory against the Philadelphia Eagles, Brinkley played his 100th career game.[4] The Titans finished with their third consecutive 9-7 season, but did not qualify for the playoffs, with Brinkley snapping for punter Brett Kern as he was named to his second consecutive Pro Bowl and was named Second-team All-Pro.
Entering the final year of his contract, Brinkley signed a multi-year contract extension with the Titans on September 6, 2019.[5] During the season, he snapped for Brett Kern as he was named to his third consecutive Pro Bowl and was named First-team All-Pro. On December 18, 2019, Brinkley was announced to be a Pro Bowl alternate for the 2020 Pro Bowl.[6] The Titans finished with a fourth consecutive 9-7 record and qualified for the playoffs, where they won upsets over the New England Patriots and the Baltimore Ravens before losing to eventual Super Bowl champions, the Kansas City Chiefs in the AFC Championship Game.
On September 29, 2020, Brinkley, along with DaQuan Jones and practice squad player Tommy Hudson were placed on the reserve/COVID-19 list by the team.[7] He was activated on October 11.[8] Brinkley was released on November 3.[9]
On November 12, 2021, Brinkley was signed to the Arizona Cardinals practice squad.[11] He was promoted to the active roster on November 20.[12] He was waived on November 22 and re-signed to the practice squad. He was promoted back to the active roster on December 4.[13] He was released on December 14.