The mechanism of the magnetic compass sense of migratory songbirds is thought to involve magneticallysensitive chemical reactions of light-induced radical pairs in cryptochrome proteins located in the birds' eyes. However it is not yet clear whether this mechanism would be sensitive enough to form the basis of a viable compass. In the present work, we report spin dynamics simulations of models of cryptochrome-based radical pairs to assess whether accumulation of nuclear spin polarization in multiple photocycles could lead to significant enhancements in the sensitivity with which the proteins respond to the direction of the geomagnetic field. Although buildup of nuclear polarization appears to offer sensitivity advantages in the more idealized model systems studied, we find that these enhancements do not carry over to conditions that more closely resemble the situation thought to exist in vivo. On the basis of these simulations, we conclude that buildup of nuclear polarization seems unlikely to be a source of significant improvements in the performance of cryptochrome-based radical pair magnetoreceptors.