As it turns out, this rumor is true! Monorails conduct safety checks every morning before they open the stations and they used to depart rom the Magic Kingdom Ticket Center and then race to Epcot to see which train could make it there first.
However, they no longer do this. A few short years ago after a long fourth of July celebration, a set of monorails tragically crashed, killing the conductor.
According to an article by the Examiner, "The NTSB estimates that the crash, which resulted in the death of monorail operator Austin Wuennenburg, resulted in $24 million in damages. Wuennenburg's mother, Christine Wuennenburg, filed suit against Disney World for reckless conduct that her attorneys contend led to the death of her son. The case was settled out of court last spring."
The practice of racing (which was not condoned or endorsed by the Disney company in any way) has been stopped. The speeds of the Monorail have been slowed, and guests are no longer allowed to ride in the front car with the conductor. (aw, man! I've been waiting 15 years to be able to ride in it and now I never will?!) Conductors emphasize, however, that the practice of racing did not endanger anyone as the trains ran parallel to one another and on separate tracks. These train cars have since been fixed and repainted from pink and purple to teal and peach. They have re-introduced them onto the track.
This is Disney's one and only monorail related fatality. It seems to have happened due to Disney's policy that driving do not need a spotter when going backwards on the track.