This is a single institution, phase I dose escalation study of weekly romiplostim post umbilical cord blood transplantation in patients who fail to achieve platelet engraftment by day +30. Engraftment is defined as a platelet count ≥ 20 x 109/L on 3 consecutive measurements without transfusion for 7 days. Romiplostim is administered at the assigned dose as 6 weekly injections beginning by day +42 post transplant. Up to 4 dose levels (4, 6, 8, and 10 mcg/kg/dose) will be evaluated with the maximum tolerated dose (MTD) of romiplostim determined by using the Continual Reassessment Method (CRM). The goal of this CRM will be to identify 1 of the 4 dose levels which corresponds to the desired maximum toxicity rate of 20% or less.