Bilateral Orthotopic Lung Transplant - Bone Marrow Transplant

Inclusion Criteria: * Subject and/or parent guardian must be able to understand and provide informed consent; * Subject fulfills criteria for United Network of Organ Sharing (UNOS) listing; * Subject must have evidence of an underlying primary immunodeficiency for which Bone Marrow Transplant (BMT) is clinically indicated. Examples of such diseases include, but are not limited to: * Severe Combined Immunodeficiency (SCID) * Combined immunodeficiency with defects in T-cell-mediated immunity, including Omenn syndrome and DiGeorge Syndrome * Severe Chronic Neutropenia * Chronic Granulomatous Disease (CGD) * Hyper Immunoglobulin E (IgE) Syndrome or Job's Syndrome * CD40 or CD40L deficiency * Wiskott-Aldrich Syndrome * Mendelian Susceptibility to Mycobacterial Disease * GATA2-associated Immunodeficiency. * Subjects must have evidence of end-stage lung disease and be candidates for bilateral orthotopic lung transplant as determined by the lung transplant team; * Glomerular filtration rate (GFR) ≥ 50 mL/min/1.73 m\^2; * Aspartate aminotransferase (AST), Alanine aminotransaminase (ALT) ≤ 4x upper limit of normal, total bilirubin ≤ 2.5 mg/dL, normal INR; * Cardiac ejection fraction ≥ 40% or shortening fraction ≥ 26%; * Negative pregnancy test for females \>10 years old or who have reached menarche, unless surgically sterilized; * All females of childbearing potential and sexually active males must agree to use a Food and Drug Administration (FDA) approved method of birth control for up to 24 months after BMT or for as long as they are taking any medication that may harm a pregnancy, an unborn child or may cause birth defect; and * Subject and/or parent guardian will also be counseled regarding the potential risks of infertility following BMT and advised to discuss sperm banking or oocyte * harvesting. Eligibility for Bone Marrow Transplant\*: * GFR \>50 mL/min/1.73 m\^2; * AST, ALT \<4x upper limit of normal, Total bilirubin \< 2.5 mg/dL; * Cardiac ejection fraction ≥40% or shortening fraction of at least 26%; * Human Immunodeficiency Virus (HIV) negative by serology and PCR; * Human T-lymphotropic virus (HTLV) serology negative; * Forced vital capacity (FVC) and Forced expiratory volume (FEV1) ≥ 40% predicted for age and SpO2 of \>90% at rest on room air AND with clearance by the lung transplant team; * Absence of uncontrolled infection as determined by blood cultures and radiographic results of previously affected sites, in particular, pulmonary densities during the past 2 weeks prior to chemotherapy; * Absence of Acute Cellular Rejection (ACR); and * Bone marrow processing has been completed, and an appropriate stem cell product is available for administration. * Note: The decision to proceed with the BMT will be at the discretion of the lung transplant team following clearance by the bone marrow team based on the criteria below. The conditioning for the BMT will begin no less than 8 weeks following the lung transplant. Exclusion Criteria: * Inability or unwillingness of a participant to give written informed consent or comply with study protocol; * Subjects who have underlying malignant conditions; * Subjects who have non-malignant conditions that do not require hematopoietic stem cell transplantation; * Human Immunodeficiency Virus (HIV) positive by serology or polymerase chain reaction (PCR), human T-lymphotropic virus (HTLV) positive by serology; * Females who are pregnant or who are lactating; * Allergy to dimethyl sulfoxide (DMSO) or any other ingredient used in the manufacturing of the stem cell product; * Uncontrolled pulmonary infection, as determined by radiographic findings and/or significant clinical deterioration. \-- Pulmonary colonization with multiple organisms is common, and will not be considered an exclusion criterion. * Uncontrolled systemic infection, as determined by the appropriate confirmatory testing e.g. blood cultures, PCR testing, etc.; * Recent recipient of any licensed or investigational live attenuated vaccine(s), within 4 weeks of transplant; or * Past or current medical problems or findings from physical examination or laboratory testing that are not listed above, which, in the opinion of the investigator, may pose: * additional risks from participation in the study, * may interfere with the participant's ability to comply with study requirements, * or that may impact the quality or interpretation of the data obtained from the study.