A Pilot Study to Evaluate the Co-Infusion of Ex Vivo Expanded Cord Blood Cells With an Unmanipulated Cord Blood Unit in Patients Undergoing Cord Blood Transplant for Hematologic Malignancies

Inclusion Criteria: * Patient has no existing 0-1 HLA-A, B, C, DRB1 and DQB1 matched related donor * Acute Myeloid Leukemia: * High risk first complete remission (CR1) as evidenced by preceding myelodysplastic syndromes (MDS), high risk cytogenetics (for example, monosomy 5 or 7, or HR as defined by referring institution treatment protocol), \>= 2 cycles to obtain complete remission (CR), erythroblastic or megakaryocytic leukemia; \>= second complete remission (CR2); * All patients must be in CR as defined by hematologic recovery and \< 5% blasts by morphology within the bone marrow and a cellularity of \>= 15%; * Patients in which adequate marrow/biopsy specimens cannot be obtained to determine remission status by morphologic assessment, but have fulfilled criteria of remission by flow cytometry (\< 5% blasts) and, recovery of peripheral blood counts with no circulating blasts, may still be eligible; reasonable attempts must be made to obtain an adequate specimen for morphologic assessment, including possible repeat procedures * Acute lymphoblastic leukemia: * High risk CR1 (for example, but not limited to: t(9;22), t(1;19), t(4;11) or other mixed-lineage leukemia (MLL) rearrangements, hypodiploid); * \> 1 cycle to obtain CR; * \>= CR2; * All patients must be in CR as defined by hematologic recovery and \< 5% blasts by morphology within the bone marrow and a cellularity of \>= 15%; * Patients in which adequate marrow/biopsy specimens can not be obtained to determine remission status by morphologic assessment, but have fulfilled criteria of remission by flow cytometry (\< 5% blasts) and, recovery of peripheral blood counts with no circulating blasts, may still be eligible; reasonable attempts must be made to obtain an adequate specimen for morphologic assessment, including possible repeat procedures * Chronic Myelogenous Leukemia: * Patients in blast crisis (BC) must receive therapy and must achieve accelerated phase (AP)/chronic phase (CP) in order to be eligible (patients who remain in BC are not eligible); * If in first chronic phase, patient must have failed or be intolerant to imatinib mesylate * Myelodysplasia (MDS): * International Prognostic Scoring System (IPSS) Int-2 or high risk (Refractory anemia with excess blasts \[RAEB\], refractory anemia with excess blasts in transformation \[RAEBt\]) or refractory anemia with severe pancytopenia or high risk cytogenetics; * Blasts must be \< 10% morphologically in representative bone marrow aspirate (obtained \< 2 weeks from enrollment) * Lymphoblastic lymphoma, Burkitt's lymphoma, and other high-grade non-Hodgkin lymphoma (NHL) after initial therapy if stage III/IV in first partial remission (PR1) or after progression if stage I/II \< 1 year; Stage III/IV patients are eligible after progression in complete response (CR)/partial response (PR) * Chronic lymphocytic leukemia/small lymphocytic lymphoma (CLL/SLL), marginal zone B-cell lymphoma, lymphoplasmacytic lymphoma or follicular lymphoma that have progressed after at least two different prior therapies; patients with bulky disease (nodal mass greater than 5 cm) should be considered for debulking chemotherapy before transplant (these patients must be presented at Patient Care Conference \[PCC\] prior to enrollment given potential competing eligibility on autotransplant protocols) * Mantle-cell lymphoma, prolymphocytic leukemia: Eligible after initial therapy in \>= CR1 or \>= PR1 * Large cell NHL \> CR2/ \> PR2: * Patients in CR2/PR2 with initial short remission (\< 6 months) are eligible; * These patients must be presented at PCC prior to enrollment given potential competing eligibility on autotransplant protocols * Multiple myeloma beyond PR2: Patients with chromosome 13 abnormalities, first response lasting less than 6 months, or beta-2 microglobulin \> 3 mg/L, may be considered for this protocol after initial therapy * Serum creatinine =\< 2.0 mg/dL (adults) and creatinine clearance \> 60 ml/min (pediatrics) * Patients with clinical or laboratory evidence of liver disease will be evaluated for the cause of liver disease, its clinical severity in terms of liver function, histology, and the degree of portal hypertension; patients with fulminant liver failure, cirrhosis with evidence of portal hypertension or bridging fibrosis, alcoholic hepatitis, esophageal varices, a history of bleeding esophageal varices, hepatic encephalopathy, or correctable hepatic synthetic dysfunction evidenced by prolongation of the prothrombin time, ascites related to portal hypertension, bacterial or fungal abscess, biliary obstruction, chronic viral hepatitis with total serum bilirubin \> 3 mg/dL and symptomatic biliary disease will be excluded * Diffusing capacity of the lung for carbon monoxide corrected (DLCOcorr) \> 50% normal * Left ventricular ejection fraction \>= 45% or shortening fraction \> 26% * Karnofsky score \>= 70% (adults) or Lansky score \>= 50% (pediatrics) Exclusion Criteria: * Acute leukemia in relapse (\>= 5% marrow blasts by morphology) * Active central nervous system (CNS) leukemia involvement at the time of study enrollment (cerebrospinal fluid with \> 5 white blood cells (WBC)/mm\^3 AND malignant cells on cytospin) * Chemotherapy refractory large cell lymphoma and high grade NHL (progressive disease after \> 2 salvage regimens) * Female patients who are pregnant or breastfeeding * Karnofsky performance status \< 70% (adults) or Lanksy score \< 50% (pediatrics) * Prior autologous or allogeneic stem cell transplant with myeloablative preparative regimen (If =\< 18 years old, prior myeloablative transplant within the last 6 months) * Uncontrolled viral, or bacterial infection at the time of study enrollment * Active or recent (prior 6 months) invasive fungal infection without ID consult and approval * Seropositive for human immunodeficiency virus (HIV) * Consenting 5 of 6 or 6 of 6 HLA-matched related donor available * Unable to provide informed consent * Use of any other experimental drug within 28 days of baseline