Coach Dave Rose announced the moves Monday afternoon.
Bartley averaged 3.9 points as a freshman and 2.5 points in 33 games during his sophomore year last season. The 6-foot-3 guard is from Baton Rouge, Louisiana.
Neilson played in 34 games and started five as a freshman in 2014-15. The 6-foot-10 forward from Mission Viejo, California, averaged 3.1 points and 2.4 rebounds.
The Cougars went 25-10 last season and advanced to the NCAA Tournament.