Moncur's most successful year came in 2001 when he won the gold medal in the 400m at the 2001 World Championships, however he has not managed to replicate this success. He also picked up a silver medal in the 4x400m relay. Moncur has only managed to make one more appearance in the individual final of a major championship, at the 2007 World Championships where he finished 8th.
More success, however, has come has part of the Bahamas relay team. At the 2005 World Championships he (together with Nathaniel McKinney, Andrae Williams and Chris Brown) won a silver medal in 4 x 400 metres relay.
Moncur's personal best time and national record in the 400m is 44.45 seconds, achieved in July 2001 in Madrid. However, Chris Brown tied Moncur's national record at the 2007 World Championships.