The Chief Flight Instructor reported that, according to the student pilot, during landing, the right crosswind increased. He added that the airplane turned to the left, touched down hard, and bounced. Following the bounce, the student tried to hold attitude, but the airplane descended, bounced a second time and the nose landing gear collapsed. During the third touchdown, the propeller struck the runway and the airplane skidded to a stop on the runway. The airplane sustained substantial damage to the engine mounts. The Chief Flight Instructor reported that there were no preaccident mechanical failures or malfunctions with the airplane that would have precluded normal operation. The automated weather observation station located on the airport reported that, about the time of the accident, the wind was from 210° at 10 knots. The airplane landed on runway 17L.