T Cells Expressing Fully-human Anti-CD19 and Anti-CD20 Chimeric Antigen Receptors for Treating B-cell Malignancies and Hodgkin Lymphoma

Background: -Cluster of differentiation 19 (CD19) and cluster of differentiation 20 (CD20) are often found on certain cancer cells. Researchers think that a person's T cells can be modified in a lab to kill cells that have CD19 and CD20 on the surface. Objective: -To see if it is safe to give anti-CD19 and anti-CD20 CAR T cells to people with a B cell cancer or Hodgkin lymphoma. Eligibility: -People ages 18 and older with a B cell cancer or Hodgkin lymphoma that has not been controlled with standard therapies Design: * Participants will be screened under protocol 01C0129 with: * Medical history * Physical exam * Blood and heart tests * Bone marrow biopsy: A needle is inserted into the participant's hip bone to remove a small amount of marrow. Scans * Participants will have apheresis: Blood will be removed through a vein. The blood with circulate through a machine that removes the T cells. The rest of the blood will be returned to the participant. * Once a day for 3 days before they get the T cells, participants will receive chemotherapy through a vein. * Participants will receive the T cells through a vein. They will stay in the hospital for at least 9 days. * Participants may have a lumbar puncture: A needle will remove fluid from the spinal cord. * Participants may have a tumor biopsy. * Participants will repeat the screening tests throughout the study. * Participants will have follow-up visits 2 weeks after infusion; monthly for 4 months; at 6, 9, and 12 months; every 6 months for 3 years; and then annually for 5 years. Participants will then be contacted annually for 15 years.