This study will evaluate pediatric patients with malignant or non-malignant blood cell disorders who are having a blood stem cell transplant depleted of T cell receptor (TCR) alfa and beta cells that comes from a partially matched family donor. The study will assess whether immune cells, called T cells, from the family donor, that are specially grown in the laboratory and given back to the patient along with the stem cell transplant can help the immune system recover faster after transplant. As a safety measure these T cells have been programmed with a self-destruct switch so that they can be destroyed if they start to react against tissues (graft versus host disease).