Safety and Feasibility of CD19 CAR T Cells Using CliniMACS Prodigy for Relapsed/Refractory CD19 Positive ALL and NHL

Eligible Diseases: Relapsed or refractory pediatric B-Cell ALL as defined by at least one of the following criteria: * Second or greater relapse OR * Any relapse after allogeneic SCT OR * Not achieving a CR after 2 cycles of standard chemotherapy regimen (including persistent MRD positive disease) OR * Not achieving a CR after 1 cycle of standard chemotherapy for relapsed leukemia (including persistent MRD positive disease) OR * Patients with Philadelphia chromosome positive (Ph+) ALL who are intolerant or have failed 3 lines of tyrosine kinase inhibitor (TKI) therapy, or if TKI therapy is contraindicated OR * Patients who meet accepted indications for allogeneic HSCT for pediatric ALL but are deemed unfit for HSCT by their treating physician are eligible for this study. This includes high risk patients in first relapse. Patients with relapsed or refractory pediatric B cell non-Hodgkin's Lymphoma as defined by: * Refractory to second-line or later lines of standard chemotherapy OR * Patients with residual disease after primary therapy and not eligible for autologous SCT OR * Any relapse after previous allogeneic or autologous SCT OR * Beyond 1st CR with relapsed or persistent disease and not eligible or appropriate for conventional allogeneic or autologous SCT Note: patients with a history of blinatumomab therapy are eligible for this study. Inclusion Criteria: * For relapsed patients, CD19 tumor expression demonstrated in bone marrow or peripheral blood by flow cytometry at most recent relapse or reconfirmed after CD19 directed therapy in ALL patients. For patients with NHL, documentation of CD19 positivity must be available from biopsy at diagnosis or most recent tumor biopsy. * Age 0 to age 30 at the time of initial diagnosis. Note: the first three subjects enrolled must be ≥16 years of age * Karnofsky (age ≥ 16 years) or Lansky (age \< 16 years) performance status ≥ 50 at screening * Patients with active CNS leukemia involvement defined as CNS-3 by CSF findings only are eligible but will have their infusion delayed until CNS disease is reduced to CNS-1 or CNS-2 by CSF findings. Patients with other forms of active CNS-3 leukemic involvement such as CNS parenchymal or ocular disease, cranial nerve involvement or significant leptomeningeal disease are eligible if there is documented evidence of disease stabilization for at least 1 month prior to CD19 CAR T cell infusion. * Meets criteria for non-hematopoietic organ function: * Renal function: Estimated glomerular filtration rate ≥60 mL/min x appropriate estimation of patient's body surface area m2/1.73m2 using the modified Schwartz formula for pediatric patients and Crockcoft Gault formula for adults. * Liver function: Total bilirubin ≤ 2 mg/dl or ≤ 2.5 x ULN for age (unless Gilbert's syndrome) and ALT and AST ≤ 5 x ULN for age (unless related to leukemic involvement) * Cardiac function: left ventricular ejection fraction ≥40% * Pulmonary function: minimum level of pulmonary reserve defined as ≤ grade 1 dyspnea and pulse oxygenation \>91% on room air * Sexually active males and females of childbearing potential must agree to use a form of contraception considered effective and medically acceptable by the Investigator. * Signed consent by parent/guardian and assent if appropriate for subjects \< 18 years of age. Signed consent by patient/subject if ≥18 years of age. Exclusion Criteria: * Acute/ongoing neurologic toxicity \> Grade 1 with the exception of a history of controlled seizures or fixed neurologic deficits that have been stable/improving over the past 1 months. * Active untreated infection. Viremia by PCR analysis is not considered an active infection but may require immediate viral prophylaxis. Patients with possible fungal infections must have had at least 2 weeks of appropriate anti-fungal therapy and be asymptomatic. * Patients with concomitant genetic syndrome: such as patients with Fanconi anemia, Kostmann syndrome, Shwachman syndrome or any other known bone marrow failure syndrome. Patients with Down Syndrome will not be excluded. * Presence of Grade 2 to 4 acute or extensive chronic graft versus-host disease (GVHD) at the time of enrollment * Patient has participated in an investigational research study using an investigational agent within the last 30 days prior to screening * Pregnant or nursing (lactating) women. * Active or latent hepatitis B or active hepatitis C (test within 8 weeks of screening), or any uncontrolled infection at screening * HIV positive test within 8 weeks of screening * Allogeneic HSCT within 3 months of enrollment * Any prior CD19 CAR T cell therapy