To learn more about the way the T cells are working in the patient's body, an extra 20-40 mL (4-8 teaspoons) of blood will be taken before each cycle of 5-aza (if applicable), 2 weeks after each cycle of 5-aza (if applicable), before each T-cell infusion, and at Weeks 1, 2, 4 and 6. One additional blood sample might be drawn 3 to 4 days post the T-cell infusion; this is optional. Afterwards, blood will be collected at 3, 6, 9 and 12 months after the last infusion. The blood may be drawn from a central line at the time of the patient's regular blood tests. Investigators will use this blood to see how long the T cells last,and to look at the immune response to the patient's cancer.