Very important: did you feed your chocobo Sylkis Greens? They're very expensive, but they raise your bird's stats. The Chocobo Sage sells them (he's in the small house in the middle of Icicle Inn mountains).
However, which kind of Chocobo are you racing?
A Good or a Great chocobo do not need to be raced past Rank B to breed a blue or a green chocobo.
Blue and green can be easily be raced to rank A, and with a gold or a black one you can reach Rank S.