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 days prior to enrollment) * Creatinine \< 2 mg/dL (or a glomerular filtration rate \> 45) (within 30-60 days prior to enrollment) * Must be willing and able to accept at least one leukapheresis procedure * Must be willing and able to provide written informed consent Exclusion Criteria: * Inability to purify \>= 1 x 10\^7 T cells from leukapheresis product * Previously known hypersensitivity to any of the agents used in this study; known sensitivity to cyclophosphamide or fludarabine * Received systemic treatment for cancer, including immunotherapy, within 14 days prior to initiation of conditioning chemotherapy administration within this protocol. Patients who have received anti-CD19 CAR T-cells will be excluded from this trial. Consistent with current trials, patients may otherwise be given bridging therapy at the discretion of the lead study investigator * Patients who have received an allograft transplant will NOT be allowed to participate in the trial. Patients who have received an autologous transplant will not be excluded and may participate in the trial * Potential requirement for systemic corticosteroids or concurrent immunosuppressive drugs based on prior history or received systemic steroids within the last 2 weeks prior to enrollment (inhaled or topical steroids at standard doses are allowed) * Human immunodeficiency virus (HIV) seropositivity or other congenital or acquired immune deficiency state, which would increase the risk of opportunistic infections and other complications during chemotherapy-induced lymphodepletion. If there is a positive result in the infectious disease testing that was not previously known, the patient will be referred to their primary physician and/or infectious disease specialist * Hepatitis B or C seropositivity with evidence of ongoing liver damage, which would increase the likelihood of hepatic toxicities from the chemotherapy conditioning regimen and supportive treatments. If there is a positive result in the infectious disease testing that was not previously known, the patient will be referred to their primary physician and/or infectious disease specialist * Dementia or significantly altered mental status that would prohibit the understanding or rendering of informed consent and compliance with the requirements of this protocol * Known clinically active brain metastases. Prior evidence of brain metastasis successfully treated with surgery or radiation therapy will not be exclusion for participation as long as they are deemed under control at the time of study enrollment and there are no neurological signs of potential brain metastases. A brain magnetic resonance imaging (MRI) scan taken within 60 days of screening may be used, otherwise a brain MRI must be performed to confirm absence of brain metastases * A Tiffeneau-Pinelli index \< 70% of the predicted value. Subjects will be excluded if pulmonary function tests indicate they have insufficient pulmonary capability * A left ventricular ejection fraction as determined by an echocardiogram lower than 40% would preclude participation * Pregnancy or breast-feeding. Female patients must be surgically sterile or be postmenopausal for two years, or must agree to use effective contraception during the period of treatment and for 6 months afterwards. All female patients with reproductive potential must have a negative pregnancy test (serum/urine) at screening and again within 14 days from starting the conditioning chemotherapy. The definition of effective contraception will be based on the judgment of the study investigators. Patients who are breastfeeding are not allowed on this study * History of other malignancy in the past 3 years with the following exceptions: * Malignancy treated with curative intent and no known active disease * Adequately treated non-melanoma skin cancer without evidence of disease * Adequately treated cervical carcinoma in situ without evidence of disease * Adequately treated breast ductile carcinoma without evidence of disease * Prostate cancer with a Gleason score less than 6 with undetectable prostate specific antigen over 12 months * Adequately treated urothelial non-invasive carcinoma or carcinoma in situ * Similar neo-plastic conditions with an expectation of greater than 95% disease free survival