Aneuploidy is a feature of many cancers. Recent studies demonstrate that in the hematopoietic stem and progenitor cell (HSPC) compartment aneuploid cells have reduced fitness and are efficiently purged from the bone marrow. However, early phases of hematopoietic reconstitution following bone marrow transplantation provide a window of opportunity whereby aneuploid cells rise in frequency, only to decline to basal levels thereafter. Here we demonstrate by Monte Carlo modeling that two mechanisms could underlie this aneuploidy peak: rapid expansion of the engrafted HSPC population and bone marrow microenvironment degradation caused by pre-transplantation radiation treatment. Both mechanisms reduce the strength of purifying selection acting in early post-transplantation bone marrow. We explore the contribution of other factors such as alterations in cell division rates that affect the strength of purifying selection, the balance of drift and selection imposed by the HSPC population size, and the mutation-selection balance dependent on the rate of aneuploidy generation per cell division. We propose a somatic evolutionary model for the dynamics of cells with aneuploidy or other fitness-reducing mutations during hematopoietic reconstitution following bone marrow transplantation. Similar alterations in the strength of purifying selection during cancer development could help explain the paradox of aneuploidy abundance in tumors despite somatic fitness costs.