This study evaluates the efficacy of abatacept in the treatment of relapsing-remitting MS. In the first phase of the study, all participants will receive 8 intravenous treatments over a period of 24 weeks. Then, if a participant remains eligible, they will enter the second phase of the study and will receive another 8 treatments over the following 24 weeks. Two-thirds (2 out of 3) of participants will receive the study drug abatacept in the first phase, and then an inactive form (placebo) of the drug in the second phase. The remaining one-third (1 in 3) will get the placebo first, then the study drug in the second phase if they remain eligible. Therefore, all participants in the ACCLAIM trial will have the opportunity to receive the study drug abatacept if they remain healthy during the study. Participants will be asked to return for a follow-up visit 12 weeks after all treatments have been completed.