Palbociclib in Combination With Chemotherapy in Treating Children With Relapsed Acute Lymphoblastic Leukemia (ALL) or Lymphoblastic Lymphoma (LL)

Inclusion Criteria: * Patients with recurrent or refractory B- or T-lineage lymphoblastic leukemia and lymphoma. * Patients with leukemia must have ≥ 5% (M2 or M3) bone marrow blasts with or without an extramedullary site of relapse. Morphologic relapse for M2 should be confirmed using flow cytometry, FISH and/or cytogenetics or molecular techniques. * Patients with LL must have either measurable or evaluable disease. * Patients with first or greater relapsed T-lineage ALL or LL and second or greater relapsed B-lineage ALL or LL are eligible. * Patients with primary refractory disease with at least 2 prior induction attempts or first relapse refractory to at least one prior re-induction attempt are eligible. * Karnofsky \>= 50% for patients \> 16 years of age and Lansky \>= 50 for patients \<= 16 years of age. * Note: Patients who are unable to walk because of paralysis, but who are up in a wheelchair, will be considered ambulatory for the purpose of assessing the performance score. * Patients must have fully recovered from the acute toxic effects of all prior anti-cancer therapy and must meet the following minimum duration from prior anti-cancer directed therapy prior to enrollment. If after the required timeframe, the numerical eligibility criteria are met, e.g., blood count criteria, the patient is considered to have recovered adequately. 1. Cytotoxic chemotherapy or other anti-cancer agents known to be myelosuppressive. See DVL homepage for commercial and Phase 1 investigational agent classifications. For agents not listed, the duration of this interval must be discussed with the study chair and the study-assigned Research Coordinator prior to enrollment. * A waiting period prior to enrollment is not required for patients receiving standard cytotoxic maintenance chemotherapy (i.e., corticosteroid, vincristine, 6MP, and/or methotrexate). * Intrathecal cytotoxic therapy: No waiting period is required for patients having received intrathecal cytarabine, methotrexate, and/or hydrocortisone. Intrathecal chemotherapy given at the time of diagnostic LP to evaluate for relapse prior to study enrollment is allowed. * At least 14 days must have elapsed after the completion of other cytotoxic therapy, with the exception of hydroxyurea, for patients not receiving standard maintenance therapy. Additionally, patients must have fully recovered from all acute toxic effects of prior therapy. * NOTE: Cytoreduction with hydroxyurea in patients can be initiated and continued for up to 24 hours prior to the start of protocol therapy. * Note: Intrathecal chemotherapy that is given up to 72 hours prior to initiation of systemic chemotherapy per AINV18P1 counts as protocol therapy and not prior anti-cancer therapy. Intrathecal chemotherapy given \> 72 hours prior does not count as protocol therapy. 2. Anti-cancer agents not known to be myelosuppressive (e.g. not associated with reduced platelet or ANC counts): \>= 7 days after the last dose of agent. See DVL homepage for commercial and Phase 1 investigational agent classifications. For agents not listed, the duration of this interval must be discussed with the study chair and the study-assigned Research Coordinator prior to enrollment. * NOTE: Cytoreduction with prednisone or methylprednisolone for \<= 120 hours (5 days) in patients can be initiated and continued for up to 24 hours prior to the start of protocol therapy. 3. Antibodies: \>= 21 days must have elapsed from infusion of last dose of antibody with the exception of blinatumomab, and toxicity related to prior antibody therapy must be recovered to Grade \<= 1. Patients must have been off blinatumomab infusion for at least 14 days and all drug related toxicity must have resolved to Grade \<= 1. 4. Corticosteroids: If used to modify immune adverse events related to prior therapy, \>= 14 days must have elapsed since last dose of corticosteroid and toxicity related to prior immune therapy must be recovered to Grade \<= 1 off corticosteroids. 5. Hematopoietic growth factors: \>= 14 days after the last dose of a long-acting growth factor (e.g. pegfilgrastim) or 7 days for short-acting growth factor. For agents that have known adverse events occurring beyond 7 days after administration, this period must be extended beyond the time during which adverse events are known to occur. The duration of this interval must be discussed with the study chair and the study-assigned Research Coordinator. 6. Interleukins, Interferons and Cytokines (other than Hematopoietic Growth Factors): \>= 21 days after the completion of interleukins, interferon or cytokines (other than Hematopoietic Growth Factors) 7. Stem cell Infusions (with or without TBI): * Allogeneic (non-autologous) bone marrow or stem cell transplant, or any stem cell infusion including DLI or boost infusion: \>= 84 days after infusion and no evidence of GVHD. * Autologous stem cell infusion including boost infusion: \>= 42 days. 8. Cellular Therapy: \>= 30 days after the completion of any type of cellular therapy (e.g. modified T cells, NK cells, dendritic cells, etc.) 9. XRT/External Beam Irradiation including Protons: \>= 14 days after local XRT; \>= 150 days after TBI, craniospinal XRT or if radiation to \>= 50% of the pelvis; \>= 42 days if other substantial BM radiation. 10. Patients must not have received prior exposure to palbociclib or another CDK4/6 inhibitor. * Adequate Renal Function Defined as: * Creatinine clearance or radioisotope GFR ≥ 70ml/min/1.73 m2 or * A serum creatinine based on age/gender as follows: * Age: 1 to \< 2 years; Male: 0.6 mg/dL; Female: 0.6 mg/dL * Age: 2 to \< 6 years; Male: 0.8 mg/dL; Female: 0.8 mg/dL * Age: 6 to \< 10 years; Male: 1 mg/dL; Female: 1 mg/dL * Age: 10 to \< 13 years; Male: 1.2 mg/dL; Female: 1.2 mg/dL * Age: 13 to \< 16 years; Male: 1.5 mg/dL; Female: 1.4 mg/dL * Age: \>= 16 years; Male: 1.7 mg/dL; Female: 1.4 mg/dL * Adequate Liver Function Defined as: * bilirubin (sum of conjugated + unconjugated) \<= 1.5 x upper limit of normal (ULN) for age * SGPT (ALT) \<= 225 U/L unless disease-related. For the purpose of this study, the ULN for SGPT is 45 U/L. * Serum albumin \>= 2 g/dL. * Adequate Cardiac Function Defined As: * Shortening fraction of \>= 27% by echocardiogram, or * Ejection fraction of \>= 50% by gated radionuclide study. * All patients and/or their parents or legally authorized representatives must sign a written informed consent. Assent, when appropriate, will be obtained according to institutional guidelines. Exclusion Criteria: * Pregnant or breast-feeding women will not be entered on this study due to risks of fetal and teratogenic adverse events as seen in animal/human studies. Based on the mechanism of action, palbociclib may be expected to cause fetal harm if used during pregnancy. Pregnancy tests must be obtained in girls who are post-menarche. Males or females of reproductive potential may not participate unless they have agreed to use an effective contraceptive method for the duration of study therapy. Women of reproductive potential should use effective contraception during treatment and for at least 3 weeks after the last dose of palbociclib. Males with female partners of reproductive potential should use effective contraception during treatment and for 3 months after the last dose of palbociclib. Animal data suggests that palbociclib may affect male fertility. * Prednisone or methylprednisolone for ≤ 120 hours (5 days) may be administered for cytoreduction up to 24 hours prior to the start of protocol therapy and as treatment for allergic reactions or for physiologic replacement/stress dosing of hydrocortisone for documented adrenal insufficiency. Corticosteroids are not allowed for other indications. If used to modify immune adverse events related to prior therapy, ≥ 14 days must have elapsed since last dose of corticosteroid. * Patients who are currently receiving another investigational drug. * Patients who are currently receiving other anti-cancer agents are not eligible \[except patients receiving hydroxyurea, which may be continued until 24 hours prior to start of protocol therapy\]. * Patients who are currently receiving drugs that are strong inhibitors and/or inducers of CYP3A4 or sensitive CYP3A4 substrates and CYP3A4 substrates with a narrow therapeutic range are not eligible. Strong inducers or inhibitors of CYP3A4 are prohibited from 14 days prior to enrollment to the end of the study. * Patients who are receiving cyclosporine, tacrolimus or other agents to prevent graft-versus-host disease post bone marrow transplant. * Patients must be able to swallow intact capsules or liquid. Patients that are unable to swallow oral medications may receive palbociclib through an NG tube. G tube administration is not allowed. * Patients who have an uncontrolled infection defined as below: * Fever above 38.2°C within 48 hours of study enrollment with clinical signs of infection. Fever that is determined to be due to tumor burden is allowed if patients have documented negative blood cultures for at least 48 hours prior to enrollment and no concurrent signs or symptoms of active infection or hemodynamic instability. * A positive fungal culture within 30 days of study enrollment or active therapy for presumed invasive fungal infection. * Patients may be receiving IV or oral antibiotics to complete a course of therapy for a prior documented infection as long as cultures have been negative for at least 48 hours and signs or symptoms of active infection have resolved. For patients with c. difficile diarrhea, at least 72 hours of antibacterial therapy must have elapsed and stools must have normalized to baseline. * Active viral or protozoal infection requiring IV treatment. * Patients known to have one of the following concomitant genetic syndromes: Down syndrome, Bloom syndrome, ataxia-telangiectasia, Fanconi anemia, Kostmann syndrome, Shwachmann syndrome or any other known bone marrow failure syndrome. * Patients who in the opinion of the investigator may not be able to comply with the safety monitoring requirements of the study. * Cumulative prior anthracycline exposure must not exceed 400 mg/m2 of DOXOrubicin equivalents