The Jamaican bobsled team came to being in the 1980s as a brainchild of George Fitch and William Maloney, two American businessmen that had business ties to Jamaica. The 1988 Olympic team was seen as a novelty as they experienced many difficulties and crashed several times. But by 1992, they had improved to the point that their 4 man team finished ahead of the United States, France, Russia, and Italy. They won the gold medal in the 2000 World Push Championships and posted the fastest starting time at the World Cup.
Jamaica has a bobsled team because there are folk there who love the sport and are good at the sport.
Look for them at the 2010 Winter Games in Vancouver.
Actually, they did not qualify for this years olympics. UPDATE from this other persons answer... not correct link to prove => The Jamaican Bobsled Team failed to qualify for both the 2006 Winter Olympics in Turin, Italy and the 2010 Winter Olympics in Vancouver, British Columbia. http://en.wikipedia.org/wiki/Jamaica_national_bobsled_team