Bilateral Orthotopic Lung Transplant - Bone Marrow Transplant

The primary purpose of this study is to evaluate the safety and efficacy of performing bilateral orthotopic lung transplantation (BOLT) followed by cadaveric, partially HLA-matched CD3+/CD19+-depleted hematopoietic stem cell transplantation (HSCT) from the same donor for participants with primary immunodeficiency diseases (PID) and end-stage lung disease. For many patients with primary immunodeficiencies, HSCT, which we refer to as Bone Marrow Transplant (BMT) is a curative, life-saving therapy, resulting in restoration of function in the immune system. Patients with primary immunodeficiencies often develop pulmonary complications as a result of chronic or recurrent infections, making them ineligible for BMT due to the high risk of mortality and pulmonary complications. Lung transplant prior to BMT would allow for restoration of pulmonary function prior to BMT, allowing PID patients to proceed to BMT , which would be curative for the patient's underlying immunodeficiency. As a secondary aim, after successful engraftment with donor bone marrow, the feasibility of participants tolerating planned withdrawal of immunosuppression and achieving eventual freedom from all immunosuppressive drugs and attaining a tolerant state will be assessed. This is a single center study in which participants receive a cadaveric, partially Human Leukocyte Antigen (HLA)-matched lung transplant followed by a CD3+/CD19+ depleted bone marrow transplant (BMT) from the same donor. In this study, the investigators will use a ≥ 2/6 HLA-matched T cell depleted bone marrow transplant from a cadaveric organ donor with an identical ABO blood type as the recipient. Participants will undergo: * Bilateral orthotopic lung transplant (BOLT) utilizing basiliximab induction or an alternate induction therapy based on their underlying disease. Rituximab may be initiated prior to the lung transplant, with tacrolimus as the ongoing maintenance immunosuppression. * BMT utilizing CD3+/CD19+-depleted bone marrow with bone marrow conditioning beginning no less than 8 weeks after BOLT. The duration of participant involvement in the trial is up to 2 years post-BMT.