The first 12 weeks of this study will compare the efficacy of etanercept 50 mg once-weekly to placebo in subjects with rheumatoid arthritis who have not responded well to infliximab or adalimumab plus methotrexate. This comparison will be performed for all subjects and separately for subjects who are anti-drug antibody positive for one of these medications. From week 12 to week 24, all subjects will receive etanercept 50 mg once-weekly. The effect of anti-drug antibody status on the efficacy of etanercept as well as the safety profile of etanercept in these subjects will also be evaluated throughout the study.