Can the Body's own immune cells beat a recurrent brain tumor?

Copied word for word from the study's registry entry, so the wording is the study team's rather than ours.

Inclusion Criteria: Screening: * Age ≤ 30 years of age. * Suspected first recurrence/progression of MB/PNET since completion of definitive focal +/- craniospinal irradiation. Disease progression prior to receiving definitive focal +/- craniospinal irradiation will not disqualify patients from enrollment if they have subsequently failed definitive radiotherapy and are at first recurrence/progression at time of enrollment. Patients who are unable to receive radiation therapy due to genetic disorders that put them at significant risk for radiation-induced secondary malignancies (i.e. Gorlin's syndrome or NF1 mutation) are eligible for enrollment at first disease recurrence/progression. Re-MATCH Protocol: * Patients must have histologically confirmed recurrent MB/PNET that is a first relapse/progression after completion of definitive radiotherapy +/- craniospinal irradiation. Patients with a first relapse/progression who are unable to receive radiation therapy due to genetic disorders that put them at significant risk for radiation-induced secondary malignancies (ie. Gorlin's syndrome or NF1 mutation) are eligible for enrollment. * Patients with neurological deficits should have deficits that are stable for a minimum of 1 week prior to registration. * Karnofsky Performance Status of ≥ 50% or Lansky Performance Score of ≥ 50. * Absolute Neutrophil Count (ANC) ≥ 1000/µl (unsupported). * Platelets ≥ 100,000/µl (unsupported). * Hemoglobin \> 8 g/dL (may be supported). * Serum creatinine ≤ upper limit of institutional normal * Bilirubin ≤ 1.5 times upper limit of normal for age. * Serum Glutamic Oxaloacetic Transaminase (ALT) ≤ 3 times institutional upper limit of normal for age. * Serum Glutamic Oxaloacetic Transaminase (AST) ≤ 3 times institutional upper limit of normal for age. * Patients of childbearing or child-fathering potential must be willing to use a medically acceptable form of birth control, which includes abstinence, while being treated on this study. * Patient or patient guardian consent to peripheral blood stem cell (PBSC) and/or bone marrow harvest following registration if PBSC or bone marrow (CD34 count of at least 2x10\^6/kg) has not been previously stored and available for use. * Signed informed consent according to institutional guidelines must be obtained prior to registration. Exclusion Criteria: * Pregnant or need to breast feed during the study period. * Active infection requiring treatment or an unexplained febrile (\> 101.5F) illness. * Known immunosuppressive disease, human immunodeficiency virus infection, or carriers of Hepatitis B or Hepatitis C virus. * Patients with active renal, cardiac (congestive cardiac failure, myocardial infarction, myocarditis), or pulmonary disease. * Patients receiving concomitant immunosuppressive agents for medical condition. * Patients who need definitive radiotherapy for treatment of recurrent MB/PNET. Focal boost radiotherapy may be delivered prior to immunotherapy if required for local control. * Patients receiving any other concurrent anticancer or investigational drug therapy. * Patients with any clinically significant unrelated systemic illness (serious infections or significant cardiac, pulmonary, hepatic or other organ dysfunction). * Patients with inability to return for follow-up visits or obtain follow-up studies required to assess toxicity to therapy.