He won three gold medals at the 1932 Summer Olympics in Los Angeles, becoming, along with Helene Madison of United States, the most successful athlete there.

At the 1928 Olympics, besides winning a silver on the horizontal bar, he finished fourth on the rings and all-around.

At the next games he won the all-around competition with a 5.7-point gap from second place and greatly helped Italy to win the team gold.

At the 1936 Olympics Neri competed with a torn arm muscle and did not complete his events.

He retired from competitions at the onset of World War II, and after the war worked as a gymnastics coach, preparing the national team for the 1952 Olympics and training his sons Romano and Giambattista.