Genetic networks show a broad-tailed distribution of the number of interaction partners per protein, which is consistent with a power-law. It has been proposed that such broad-tailed distributions are observed because they confer robustness against mutations to the network. We evaluate this hypothesis for two genetic networks, that of the E. coli core intermediary metabolism and that of the yeast protein-interaction network. Specifically, we test the hypothesis through one of its key predictions: highly connected proteins should be more important to the cell and, thus, subject to more severe selective and evolutionary constraints. We find, however, that no correlation between highly connected proteins and evolutionary rate exists in the E. coli metabolic network and that there is only a weak correlation in the yeast protein-interaction network. Furthermore, we show that the observed correlation is function-specific within the protein-interaction network: only genes involved in the cell cycle and transcription show significant correlations. Our work sheds light on conflicting results by previous researchers by comparing data from multiple types of protein-interaction datasets and by using a closely related species as a reference taxon. The finding that highly connected proteins can tolerate just as many amino acid substitutions as other proteins leads us to conclude that power-laws in cellular networks do not reflect selection for mutational robustness.