The movie described as a 16-year-old quitting an elite gymnastics program is Raising the Bar. The 2016 Australian-American sports drama follows Kelly Johnson, a teenage gymnast who leaves her elite U.S. program after moving to Australia with her mother.
Kelly initially swears off gymnastics, but she becomes involved again when she helps her new friend compete. The story centers on her struggle with competition, friendship, relocation, and her return to the sport.
The wording calling the film “based on a true story” is inaccurate or unsupported: Raising the Bar is presented as a fictional sports drama rather than a biographical film. Full Out is a different film based on gymnast Ariana Berlin, whose career was interrupted by a serious car accident before her comeback.
Thus, the stored answer does not fit the plot clue. Raising the Bar is the best canonical title for the described movie, with the true-story characterization requiring correction in the question itself.