We report on an extensive numerical investigation of the Kardar-Parisi-Zhang equation describing nonequilibrium interfaces. Attention is paid to the dependence of the growth exponent ß on the details of the distribution of noise p(?). All distributions considered are delta correlated in space and time, and have finite cumulants. We find that ß becomes progressively more sensitive to details of the distribution with increasing dimensionality. We discuss the implications of these results for the universality hypothesis.