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.