Jozef Stumpel played his junior hockey in Czechoslovakia and represented his country at the World Junior Championships in 1991. He won a bronze medal and was selected by the Boston Bruins in that year's NHL Entry Draft. He played the majority of the next season in Germany but made his NHL debut with Boston and split the next year with the Bruins and their AHL affiliate in Providence.
He split the next two seasons once again before playing his first full year in the NHL in 1995-96. Stumpel once again represented his country in 1996 and 1997 at both the World Cup and World Championships respectively. Before the 1997-98 season began, Stumpel was traded to the Los Angeles Kings where he blossomed into a talented player.
Stumpel went on to play parts of five seasons with the Kings before he was re-acquired by the Bruins in the early stages of the 2001-02 season. A member of Slovakia's Olympic and World Championship teams in 2002, Stumpel continued providing offense for the Bruins, notching his fourth straight 50-plus point season in 2002-03 and reaching the 500-point plateau.
In the summer of 2003 Stumpel signed as a free-agent with the Los Angeles Kings, marking his second stint with the club. However, injuries limited the talented Slovak to 64 regular season games and his lowest point total (8-29-37)since the 1998-99 season. After failing to qualify for the post season, Stumpel made his fourth appearance at the World Championships in 2004 and later that summer represented his homeland at the 2004 World Cup of Hockey.
With the 2004-05 NHL season canceled due to the labour dispute, Stumpel would make himself available yet again for the 2005 World Championships held in Austria and later that summer was acquired by the Florida Panthers.
In 2006 Stumpel represented his homeland for a second time in the 2006 Winter Olympics.