This study will test the hypothesis that continuous positive airway pressure (CPAP) has an acute lowering-BP effect in autonomic failure patients with supine hypertension. CPAP is a widely-used treatment for sleep-related breathing disorders including sleep apnea, that uses mild air pressure to keep the breathing airways open. It involves using a CPAP machine that blows air into a tube connected to a mask placed over the nose, or nose and mouth. For these studies, a commercial CPAP device will be used to apply pressure sequentially at 0, 4, 8, 12 and 16 cm H2O for 1-20 minute each. Depending on the BP response and tolerability to CPAP, CPAP may be applied during the night using a CPAP level that was tolerable and showed a BP-lowering effect during the acute test.