Can a student change status during "grace" period of 30 days (J1 visa) or 60 days (F1 visa)?

Following completion of studies or completion of practical training, students have either 30 days (J1 summer work and travel and other categories) or 60 days (F1 visa holders) to leave the country. This time is not for changing status. Students are permitted to travel in the country during these days.