The purpose of this study is to evaluate the efficacy and safety of switching rheumatoid arthritis (RA) participants who have an inadequate response to their current treatment with either etanercept + methotrexate or adalimumab + methotrexate to treatment with golimumab 50 milligram (mg) subcutaneous (SC) injection (a needle inserted under the skin in the back of upper arm, upper thigh or stomach area) every 4 weeks + methotrexate. This study is also designed to evaluate the benefit and safety of switching participants from treatment with golimumab 50 mg subcutaneous injection every 4 weeks + methotrexate to golimumab 2 milligram per kilogram (mg/kg) intravenous every 8 weeks + methotrexate, for those who do not achieve a marked improvement of their RA at Week 16.