Bolden, who also played 48 snaps on defense last season as a reserve safety, averaged 22.8 yards per return on kickoffs and averaged 24.6 yards per punt return.
He took a punt back 83-yards for a touchdown against the Indianapolis Colts.
Bolden missed seven games in the 2015 regular season because of a hamstring injury.
A fourth-round pick out of Arizona State in 2012, Bolden appeared in 56 games in four seasons with the Broncos. He is the third player signed with connections to John Fox, joining linebacker Danny Trevathan and guard/center Manny Ramirez.