Laboratory Treated T Cells in Treating Patients With Relapsed or Refractory Chronic Lymphocytic Leukemia, Non-Hodgkin Lymphoma, or Acute Lymphoblastic Leukemia

Inclusion Criteria: INCLUSIONS FOR SCREENING AND LEUKAPHERESIS * Patients with CD19 expressing acute lymphoblastic leukemia (ALL), chronic lymphocytic leukemia (CLL) or non-Hodgkin lymphoma (NHL) * Ability to understand and provide informed consent * Not human immunodeficiency virus (HIV) infected INCLUSIONS FOR CAR-T CELL THERAPY * Patients with: * CLL who are beyond first remission and who have failed combination chemoimmunotherapy with regimens containing a purine analogue and anti-CD20 antibody or who were not eligible for such therapy; patients with CLL for whom ibrutinib is now standard first line therapy, must have progressed on ibrutinib; patients with fludarabine refractory disease are eligible; patients may be treated following allogeneic hematopoietic cell transplant (HCT); for the concurrent ibrutinib cohort, patients must agree to continue on or be restarted on ibrutinib and must not have had prior intolerance to ibrutinib that would prevent this; patients managed with prior dose reductions for toxicity will continue at the reduced dose for the remainder of this study * Indolent NHL or mantle cell NHL who are beyond first remission and previously treated with chemoimmunotherapy or who were not eligible for such therapy; patients who have relapsed following autologous or allogeneic HCT are eligible * Aggressive NHL such as diffuse large B-cell lymphoma (DLBCL), who have relapsed or have residual disease following treatment with curative intent; patients should have relapsed following, or not be eligible for high-dose therapy and autologous HCT; patients with chemotherapy refractory disease or marrow involvement or comorbidities precluding successful autologous HCT are eligible; patients may be treated following allogeneic HCT * Patients with CD19 expressing, relapsed or refractory ALL * Patients with one of the above diagnoses whose disease state does not qualify but who have prognostic indicators that suggest a high risk of progression of disease may be screened and undergo leukapheresis; enrollment for T cell therapy would require meeting the full disease state eligibility * Confirmation of diagnosis * Evidence of CD19 expression by immunohistochemistry or flow cytometry on any prior or current tumor specimen or high likelihood of CD19 expression based on disease histology * Karnofsky performance status \>= 60% * All patients of childbearing potential must be willing to use a contraceptive method before, during, and for at least two months after the T cell infusion * Ability to understand and provide informed consent Exclusion Criteria: EXCLUSIONS FOR CAR-T CELL THERAPY * Patients requiring ongoing daily corticosteroid therapy at a dose of \> 15 mg of prednisone per day (or equivalent); pulsed corticosteroid use for disease control is acceptable * Active autoimmune disease requiring immunosuppressive therapy is excluded unless discussed with the Principal Investigator (PI) * Serum creatinine \> 2.5 mg/dL * Serum glutamic oxaloacetic transaminase (SGOT) \> 5 x upper limit of normal * Bilirubin \> 3.0 mg/dL * Patients with clinically significant pulmonary dysfunction, as determined by medical history and physical exam should undergo pulmonary function testing; those with a forced expiratory volume in one second (FEV1) of \< 50 % of predicted will be excluded * Diffusing capacity of the lung for carbon monoxide (DLCO) (corrected) \< 40% will be excluded * Significant cardiovascular abnormalities as defined by any one of the following: New York Heart Association (NYHA) class III or IV congestive heart failure, clinically significant hypotension, uncontrolled symptomatic coronary artery disease, or a documented ejection fraction of \< 35% * Uncontrolled active infection