Participants will undergo a complete history and physical examination. A bone marrow aspiration and biopsy will be performed to confirm the type of bone marrow failure. About 5 tablespoons of blood will be drawn for baseline tests and research purposes. Daclizumab will be administered every 2 weeks by vein in a 30-minute infusion. The first dose will be given at NIH and the next four may be given at NIH or by the participant's primary hematologist. The treatment will last 8 weeks. Participants must also see their referring physician or NIH physicians every 2 weeks for blood counts. In the fourth and eighth weeks of the study and at the 3-month follow-up visit, 2 tablespoons of blood will be drawn at NIH. At the 1-month follow-up visit to NIH, 5 tablespoons of blood will be drawn and another bone marrow aspiration and biopsy performed.