Modified Immune Cells (CD19/CD20 CAR-T Cells) in Treating Patients With Recurrent or Refractory B-Cell Lymphoma or Chronic Lymphocytic Leukemia

Inclusion Criteria: * Diffuse large B-cell lymphoma (DLBCL), primary mediastinal large B-cell lymphoma (PMBCL), mantle cell lymphoma (MCL), follicular lymphoma (FL), chronic lymphocytic leukemia (CLL), or small lymphocytic lymphoma (SLL) that is refractory to standard-of-care options * DLBCL and PMBCL: primary refractory; relapsed after two prior lines of therapy * MCL, FL, CLL, and SLL: primary refractory; relapsed after three or more prior rounds of therapy * \> 30% positivity in malignant cells of either CD19 and/or CD20 * Minimum tumor burden of 1.5 cm\^3 for lymphoma * Eastern Cooperative Oncology Group (ECOG) performance status 0 or 1 * Adequate bone marrow and major organ function to undergo a T cell transplant determined within 30?60 days prior to enrollment using standard phase I criteria for organ function. Blood may be evaluated while a patient is receiving growth factor support. Patients will be re-evaluated for organ function within 14 days of beginning conditioning chemotherapy * Absolute neutrophil count (ANC) \>= 1 x 10\^9 cells/L (within 30-60 days prior to enrollment) * Platelets \>= 75 x 10\^9/L (within 30-60 days prior to enrollment) * Hemoglobin \>= 8 g/dL (with or without transfusion) (within 30-60 days prior to enrollment) * Aspartate and alanine aminotransferases (AST, ALT) =\< 2.5 x upper limit of normal (ULN) (within 30-60 days prior to enrollment) * Total bilirubin =\< 2 x ULN (except patients with documented Gilbert?s syndrome) (within 30-60 …