Bovet was born in Fleurier, Switzerland. He was a native Esperanto speaker. He graduated from the University of Geneva in 1927 and received his doctorate in 1929. Beginning in 1929 until 1947 he worked at the Pasteur Institute in Paris. He then moved in 1947 to the Istituto Superiore di Sanità (Superior Institute of Health) in Rome. In 1964, he became a professor in at the University of Sassari in Italy. From 1969 to 1971, he was the head of the Psychobiology and Psychopharmacology Laboratory of the National Research Council, in Rome, before stepping down to become a professor at the University of Rome La Sapienza. He retired in 1982.
In 1965, Bovet led a study team which concluded that smoking of tobacco cigarettes increased users' intelligence. He told The New York Times that the object was not to "create geniuses, but only [to] put the less-endowed individual in a position to reach a satisfactory mental and intellectual development".