Born in Newcastle upon Tyne, Tyne and Wear, Bartlett started his career at Newcastle United at the age of nine, where he progressed to reserve team level but never played a competitive first team match. He travelled with the first team squad for a UEFA Champions League match against Internazionale where he was an unused substitute, and was also an Academy goalkeeping coach for the Magpies.
He made his senior footballing debut at Blyth Spartans, initially as cover for the injured Craig Turns, but soon became the club’s first choice goalkeeper.
Bartlett’s impressive performances at Blyth Spartans led to an England C call up and upon his departure from the club he was inducted into Blyth Spartans’ Hall of Fame.
In July 2008 Bartlett’s contract with Blyth expired and he was signed by Kidderminster Harriers where he was an ever-present in the Conference National during the 2008–09 season. Bartlett also appeared on the losing side in the 2009 Conference National play-off Final after he signed for Cambridge United on an emergency loan. He was named as the Conference Goalkeeper of the Year for 2008–09.
On 22 June 2009 Bartlett signed for Hereford United. He helped Hereford upset League One Charlton in a first round League Cup tie on 11 August 2009 in which he saved a penalty from Charlton’s Andy Gray in normal time. In March 2010 he extended his Hereford contract for a further year. In May 2011 he signed a new two-year contract with the club.
On 8 June 2012, Bartlett signed for Gateshead. He made his debut on 11 August 2012 in a 2–2 draw with Luton Town.
Adam joined Hartlepool United in July 2015. He made his Pools debut against Morecambe FC, keeping a clean sheet in a 2-0 win. After the arrival of Joe Fryer on loan from Middlesbrough, Adam had his contract cancelled before joining Darlington.
Adam joined non-league Darlington in March 2017. The part-time role suited Bartlett who also took up the role of Middlesbrough academy goalkeeper coach.