Susceptibility to common human diseases such as cancer is influenced by many genetic and environmental factors that work together in a complex manner. The state of the art is to perform a genome-wide association study (GWAS) that measures millions of single-nucleotide polymorphisms (SNPs) throughout the genome followed by a one-SNP-at-a-time statistical analysis to detect univariate associations. This approach has identified thousands of genetic risk factors for hundreds of diseases. However, the genetic risk factors detected have very small effect sizes and collectively explain very little of the overall heritability of the disease. Nonetheless, it is assumed that the genetic component of risk is due to many independent risk factors that contribute additively. The fact that many genetic risk factors with small effects can be detected is taken as evidence to support this notion. It is our working hypothesis that the genetic architecture of common diseases is partly driven by non-additive interactions. To test this hypothesis, we developed a heuristic simulation-based method for conducting experiments about the complexity of genetic architecture. We show that a genetic architecture driven by complex interactions is highly consistent with the magnitude and distribution of univariate effects seen in real data. We compare our results with measures of univariate and interaction effects from two large-scale GWASs of sporadic breast cancer and find evidence to support our hypothesis that is consistent with the results of our computational experiment.