Neri took swimming, running, weightlifting and boxing before changing to gymnastics. In 1926 he won the national championships in the parallel bars, followed by four all-round titles in 1928–1930 and 1933. 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. He also won gold on the parallel bars and finished fourth on the floor. 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.
Neri was the first gold medalist from Rimini, and the football stadium there, Stadio Romeo Neri, bears his name.