Genome-wide association studies have discovered common genetic variants associated with susceptibility for several complex diseases, but they have been unfruitful for many others. Typically, analysis is done "agnostically," by considering one single nucleotide polymorphism (SNP) at a time and controlling the overall type I error rate by correcting for multiple testing. Such one-at-a-time analyses may be inadequate for screening genes under realistic causal models. We use oral clefting as a disease model to develop a range of toy example scenarios: risk might involve only genes, or genes and exposure, or genes, exposure, and their supermultiplicative interaction. These examples illustrate how dramatically important genetic variants can be obscured by a one-SNP-at-a-time analysis when multiple biologic pathways and multiple genes jointly influence etiology. These examples highlight the need for better methods for gene-by-environment and gene-by-gene analyses.