This phase I trial studies the side effects and best dose of CD19/CD20 chimeric antigen receptor (CAR) T-cells when given together with chemotherapy, and to see how effective they are in treating patients with non-Hodgkin's B-cell lymphoma or chronic lymphocytic leukemia that has come back (recurrent) or has not responded to treatment (refractory). In CAR-T cell therapy, a patient's white blood cells (T cells) are changed in the laboratory to produce an engineered receptor that allows the T cell to recognize and respond to CD19 and CD20 proteins. CD19 and CD20 are commonly found on non-Hodgkin?s B-cell lymphoma and chronic lymphocytic leukemia cells. Chemotherapy drugs such as fludarabine phosphate and cyclophosphamide can control cancer cells by killing them, by preventing their growth, or by stopping them from spreading. Combining CD19/CD20 CAR-T cells and chemotherapy may help treat patients with recurrent or refractory B-cell lymphoma or chronic lymphocytic leukemia.

PRIMARY OBJECTIVES: I. To evaluate the safety of the autologous anti-CD19/anti-CD20 CAR-expressing naive/memory T cells (CART19/20), including determination of the maximum tolerated dose and assessment for replication competent lentivirus (RCL). SECONDARY OBJECTIVES: I. Clinical response. Ia. Overall response rate. Ib. Duration of remission. Ic. Progression-free survival. Id. Overall survival. II. CD19/CD20 bispecific CAR transgenic T-cell persistence. IIa. T-cell monitoring and analyses. IIb. Evidence of B-cell aplasia. EXPLORATORY OBJECTIVES: I. To determine the serum levels of cytokines associated with cytokine release syndrome (CRS) in subjects exhibiting \> grade-2 CRS following CART19/20 cell treatment. OUTLINE: This is a dose-escalation study of CD19/CD20 CAR-T cells. CONDITIONING CHEMOTHERAPY: Patients receive fludarabine phosphate intravenously (IV) over 30 minutes and cyclophosphamide IV over 60 minutes 5, 4, and 3 days before cell infusion. T-CELL INFUSION: Patients receive CD19/CD20 CAR-T cells IV on day 0. Patients with cytokine release syndrome may also receive tocilizumab IV on day 2 at the discretion of the clinical investigator. After completion of study treatment, patients are followed up daily for 14 days, on days 30, 45, 60, 70, 90, and 120, every 3 months for 2 years, every 6 months for 3 years, and then annually for a minimum of 15 years.