The purpose of this clinical research study is to determine whether abatacept treatment on a background of Disease Modifying Antirheumatic Drugs (DMARDs) will relieve the symptoms of rheumatoid arthritis (RA) in participants who are currently receiving anti-tumor necrosis factor (TNF) therapy for at least 3 months and are not responding or have taken anti-TNF therapy in the last 3 months and did not respond. The safety of treatment with abatacept will also be evaluated. This study also has a 4.5-year long-term extension beginning 6 months after the start of the study.