Bradbury left Halifax this summer after two years at the club in which he became a key player in defence, displaying versatility in playing in a back three and a back four, as well as at left-back.
"Tom is a left-sided central defender who can also play in the middle of a back three or four," Cheltenham's Director of Football Micky Moore told the club's website.
"He is a footballing centre-back with good experience and a couple of hundred first team appearances to his name.
"He is a leader, which we are also looking for having lost Will Boyle this summer.
"Tom was very sought after by teams in League One and League Two so we are very pleased to have clinched this deal."