This study will test whether the immune-suppressing drug rituximab can increase blood counts and reduce the need for transfusions in patients with moderate aplastic anemia, pure red cell aplasia, or Diamond Blackfan anemia. These are rare and serious blood disorders in which the immune system turns against bone marrow cells, causing the bone marrow to stop producing red blood cells in patients with pure red cell aplasia and Diamond Blackfan anemia, and red blood cells, white blood cells and platelets in patients with aplastic anemia. Rituximab is a laboratory-made monoclonal antibody that recognizes and destroys white blood cells called lymphocytes that are responsible for destroying bone marrow cells in these diseases. The drug is currently approved by the Food and Drug Administration for treating patients with B-cell non-Hodgkin lymphoma, a disease of white blood cells.