The purpose of this study is to test whether abatacept, a drug approved by the Food and Drug Administration to treat rheumatoid arthritis, may help blood pressure medications to work better. This will be studied in people with high blood pressure that is not well controlled on three or more blood pressure medications, the condition also known as resistant hypertension. We expect to show that adding abatacept therapy to standardized treatment of resistant hypertension will result in a greater decrease in blood pressure at 24 weeks compared to treatment with placebo and conventional blood pressure treatment.