According to the student pilot, his first two attempts to land resulted in bounced landings, and he performed a go-around each time. On the third landing attempt, the airplane approached the runway at a “shallower angle,” bounced, and impacted the nose landing gear which caused it to collapse. A postaccident examination revealed substantial damage to the engine firewall. A Federal Aviation Administration inspector reported no mechanical malfunctions or anomalies with the airplane that would have precluded normal operation.