The trained customs officer was able to place himself in the top ten in a World Cup race for the first time in December 1980 as a 19-year-old . This remained his only top result for over two years, in which injuries and form fluctuations influenced his athletic performance. It was not until January 1983 that he achieved a single-digit placement again in the Kranjska Gora slalom . In the following races his results stabilized.
When he made his Olympic debut in Sarajevo in 1984 , he achieved the greatest success of his career. Although he had not yet achieved a single World Cup podium, his coaches Georges Coquillard and Jean Béranger gave him their trust and let him go to the start. In the first run, in which his teammates Michel Canac , Yves Tavernier and Michel Vion all retired, Bouvet finished 5th. In the second run, he improved two places with a daring run and took the bronze medal behind the two brothers Phil and Steve Mahre . The first Olympic medal for the French men's team in 16 years.
Another phase of moderate results followed and in the spring of 1985 he had to wear a plaster corset due to a damaged spinal disc. It was not until almost two years after his Olympic success that he achieved the only World Cup victory of his career in the slalom in Parpan, Switzerland , the first by a Frenchman since Jean-Noël Augert in 1973. That season he achieved seventh place in the overall slalom ranking of the World Cup. After the 1989/90 season he retired from top-class sport. From 1982 to 1988 Bouvet had won five French championship titles, four of them in slalom and one in giant slalom.