Many complex networks such as computer and social networks exhibit modular structures, where links between nodes are much denser within modules than between modules. It is widely believed that cellular networks are also modular, reflecting the relative independence and coherence of different functional units in a cell. While many authors have claimed that observations from the yeast protein-protein interaction (PPI) network support the above hypothesis, the observed structural modularity may be an artifact because the current PPI data include interactions inferred from protein complexes through approaches that create modules (e.g., assigning pairwise interactions among all proteins in a complex). Here we analyze the yeast PPI network including protein complexes (PIC network) and excluding complexes (PEC network). We find that both PIC and PEC networks show a significantly greater structural modularity than that of randomly rewired networks. Nonetheless, there is little evidence that the structural modules correspond to functional units, particularly in the PEC network. More disturbingly, there is no evolutionary conservation among yeast, fly, and nematode modules at either the whole-module or protein-pair level. Neither is there a correlation between the evolutionary or phylogenetic conservation of a protein and the extent of its participation in various modules. Using computer simulation, we demonstrate that a higher-than-expected modularity can arise during network growth through a simple model of gene duplication, without natural selection for modularity. Taken together, our results suggest the intriguing possibility that the structural modules in the PPI network originated as an evolutionary byproduct without biological significance.