CD19/CD22 Chimeric Antigen Receptor (CAR) T Cells With or Without NKTR-255 in Adults With Recurrent or Refractory B Cell Malignancies

For B acute lymphoblastic leukemia (ALL) 1. Confirmed diagnosis of relapsed or refractory B-cell ALL of one of the following types: * Chemotherapy refractory disease in subjects with B-ALL, defined as progression or stable disease after one line of therapy. * Recurrence of disease after achieving CR. 2. Subjects with persistent or relapsed minimal residual disease (MRD) (by flow cytometry, PCR, FISH, or next generation sequencing) require verification of MRD positivity on two occasions at least 4 weeks apart. 3. Subjects with Philadelphia Chromosome positive acute lymphoblastic leukemia (Ph+ALL) subjects are eligible if they progressed after receiving a tyrosine kinase inhibitor (TKI). 4. Subjects with recurrence of isolated CNS relapse after achieving complete remission (CR); if relapsed with MRD, will require verification of MRD positivity on two occasions at least 4 weeks apart. 5. CD19 positive expression- CD19 expression is required at any time since diagnosis. If patient has received anti-CD19 targeted therapy (i.e. Blinatumomab or CD19-CAR T cells), then CD19 expression must be subsequently demonstrated. CD19 expression may be detected by immunohistochemistry or by flow cytometry. The choice of whether to use flow cytometry or immunohistochemistry will be determined by what is the most easily available tissue sample in each subject. In general, immunohistochemistry will be used for lymph node biopsies, flow cytometry will be used for peripheral blood and bone marrow samples. 6. Subjects who have undergone autologous SCT with disease progression or relapse following SCT are eligible. Subjects who have undergone allogeneic SCT will be eligible if, in addition to meeting other eligibility criteria, they have elelino evidence of GVHD and have been without immunosuppressive agents for at least 30 days. 7. Subjects who have undergone prior anti-CD19 or anti-CD22 CAR therapy must be at least 30 days post CAR infusion and may not have eficence of persistnce of CAR T cells in blood smples (circulating levels of genetically modified cels of \>/= 5% by flow cytometry. 8. Must have evaluable or measurable disease. Lesions that have been previously irradiated will be considered measurable only if progression has been documented following completion of radiation therapy. 9. At least 2 weeks or 5 half-lives, whichever is shorter, must have elapsed since any prior systemic therapy at the time the subject is planned for leukapheresis, except for systemic inhibitory/stimulatory immune checkpoint therapy, which requires 5 half-lives. Exceptions: 1. There is no time restriction with regard to prior intrathecal chemotherapy (incl. steroids) provided there is complete recovery from any acute toxic effects; 2. Subjects who are on standard ALL maintenance type chemotherapy (vincristine, 6-mercaptopurine or oral methotrexate) may be enrolled provided that chemotherapy is discontinued at least 1 week or 5 half-lives (whichever is shorter) prior to apheresis. 3. Subjects receiving steroid therapy at physiologic replacement doses (≤5 mg/day of prednisone or equivalent doses of other corticosteroids) only are allowed provided there has been no increase in dose for at least 2 weeks prior to starting apheresis; 4. For radiation therapy: Radiation therapy must have been completed at least 3 weeks prior to apheresis, with the exception that there is no time restriction if the volume of bone marrow treated is less than 10% and also the subject has measurable/evaluable disease outside the radiation port. 10. Toxicities due to prior therapy must be stable and recovered to ≤ Grade 1 (except for clinically non-significant toxicities such as alopecia) 11. Age 18 or older 12. Eastern cooperative oncology group (ECOG) performance status of 0, 1, or 2; or Karnofsky ≥ 60% 13. Normal Organ and Marrow Function (supportive care is allowed per institutional standards, i.e. filgrastim, transfusion) 1. ANC ≥ 1000/uL\* 2. Platelet count ≥ 50,000/uL\* 3. Absolute lymphocyte count ≥ 300/uL\* 4. Adequate renal, hepatic, pulmonary and cardiac function defined as: 5. Creatinine ≤ 2 mg/dL or creatinine clearance ≥ 60 mL/min 6. Serum ALT or AST ≤ 5x ULN (Elevated ALT/AST associated with leukemia or lymphoma involvement of the liver will not disqualify a subject; only one value required for eligibility). 7. Total bilirubin ≤ 1.5 mg/dl, except in subjects with Gilbert's syndrome. 8. Cardiac ejection fraction ≥ 45%, no evidence of physiologically significant pericardial effusion as determined by an ECHO, MUGA or Cardiac MRI \[performed within 180 days or after most recent anthracycline based treatment or mediastinal radiation therapy (whichever is most recent)\] 9. No clinically significant ECG findings 10. No clinically significant pleural effusion 11. Baseline oxygen saturation \> 92% on room air \* A subject will not be excluded because of cytopenia if it is felt by the investigator to be due to underlying leukemia/lymphoma. 14. Subjects with CNS involvement are eligible as long as there are no overt signs or symptoms that in the evaluation of the investigator would mask or interfere with the neurological assessment of toxicity. 15. Females of childbearing potential must have a negative serum or urine pregnancy test (females who have undergone surgical sterilization or who have been postmenopausal for at least 2 years are not considered to be of childbearing potential) 16. Subjects of child-bearing or child-fathering potential must be willing to practice birth control from the time of enrollment on this study and for four (4) months after receiving the preparative lymphodepletion regimen or 1 month after the last dose of NKTR\_255, whichever is later. 17. Ability to give informed consent. Must be able to give informed consent. Subjects unable to give informed consent will not be eligible for this study. =ELIGIBILITY TO RECEIVE NKTR-255= * Received a CD19/CD22 CAR-T infusion * No persisting grade ≥1 CRS or greater than grade 1 fever within 12 hours preceding NKTR-255 infusion * No grade 4 CRS within 96 hours preceding NKTR-255 infusion * No persisting grade ≥ 2 neurotoxicity on the day of NKTR-255 infusion * No previous grade ≥ 3 neurotoxicity of \> 48 hours duration at any time preceding NKTR-255 infusion * ANC ≥ 1000/µL * No intervention with tocilizumab and/or dexamethasone within 48 hours preceding NKTR-255 infusion * No active, serious, and uncontrolled infection(s) * No contraindications according to the PI's assessment * Life expectancy \> 30 days Exclusion Criteria: 1. History of other malignancy, unless disease free for at least 3 years. At the discretion of the Principal Investigator, subjects in remission for 1-2 years prior to enrollment may be deemed eligible after considering the nature of other malignancy, likelihood of recurrence during one year following CAR therapy, and impact of prior treatment on risk of CD19/CD22-CAR T cells. Subjects in remission \<1 year are not eligible. * Exception: Nonmelanoma skin cancer or carcinoma in situ (e.g. cervix, bladder, breast) is eligible. * Hormonal therapy in subjects in remission \>1 year will be allowed. 2. Presence of fungal, bacterial, viral, or other infection that is uncontrolled. Simple UTI and uncomplicated bacterial pharyngitis are permitted if responding to active treatment. 3. Known history of infection with any of the following: * HIV * Hepatitis B (HBsAg positive) * Hepatitis C virus (anti-HCV positive) A history of hepatitis B or hepatitis C is permitted if the viral load is undetectable per quantitative PCR and/or nucleic acid testing. 4. Presence of a seizure disorder, cerebrovascular ischemia/hemorrhage, dementia, cerebellar disease, or any autoimmune disease with CNS involvement that in the judgment of the investigator may impair the ability to evaluate neurotoxicity. 5. History of myocardial infarction, cardiac angioplasty or stenting, unstable angina, or other clinically significant cardiac disease within 12 months of enrollment 6. Any medical condition that in the judgement of the investigator is likely to interfere with assessment of safety or efficacy of study treatment 7. History of severe immediate hypersensitivity reaction to any of the agents used in this study 8. Women who are pregnant or breastfeeding 9. In the investigator's judgment, the subject is unlikely to complete all protocol-required study visits or procedures, including follow-up visits, or comply with the study requirements for participation. 10. Previous treatment with interleukin-2 or interleukin-15. 11. Confirmed diagnosis of relapsed/refractory biphenotypic BT cell ALL 12. Primary immunodeficiency or history of autoimmune disease (e.g. Crohns, rheumatoid arthritis, systemic lupus) requiring systemic immunosuppression/systemic disease modifying agents within the last 2 years