I think you mean angular play. It has to have axial play for the quill to go up and down (unless the motor and pulleys ride up/down with the chuck.) Drilling won't cause chatter, because the cutting action is pretty continuous. But, milling will definitely cause chatter when a cutting tooth is not buried in the work all the time. The worst is when the spindle is springy in the torsion mode and winds up during the cut, then jumps ahead when the tooth pops free of the work. I had a Bridgeport M head mill that had that problem. I put brass shims on the sides of the pulley splines, in balanced pairs, on what would normally be the "coasting" faces, as opposed to the "driving" faces, to take up the slack. It was amazing how much quieter it ran!
Jon