This is a single arm dose finding study evaluating the safety and efficacy of a BPX 501 infusion (T cells genetically modified with the inducible Caspase 9 suicide gene) of 3x10E6 to 1X10E7 cells/kg followed by an AP1903 infusion on day 7 after a partially mismatched, related, T cell-depleted hematopoietic cell transplantation (HCT) in patients with non-malignant diseases. The purpose of this clinical trial is to determine the dose of BPX 501 T cell infusion with subsequent planned infusion of AP1903 which can facilitate engraftment and prevent the occurrence of GVHD.