Inspired by the discovery of Oddershede et al. [Phys. Rev. Lett. 71, 3107 (1993)] concerning the dimensionality dependence of mass distributions after shock fragmentation, we have performed further experiments using thick plates of dry clay. We observe mass distributions with composite power laws, i.e., different exponents for fragments larger and smaller than the plate thickness. This implies that the dominant exponent for any given fragment mass corresponds to the dimensionality of the original object on the length scale of the fragment considered. We study two profoundly different models, both of which agree qualitatively with the observed features. Thus the measured mass distribution tells little about the mechanisms of the fragmentation process.