The expression dynamics of interacting genes depends, in part, on the structure of regulatory networks. Genetic regulatory networks include an overrepresentation of subgraphs commonly known as network motifs. In this article, we demonstrate that gene copy number is an omnipresent parameter that can dramatically modify the dynamical function of network motifs. We consider positive feedback, bistable feedback, and toggle switch motifs and show that variation in gene copy number, on the order of a single or few copies, can lead to multiple orders of magnitude change in gene expression and, in some cases, switches in deterministic control. Further, small changes in gene copy number for a 3-gene motif with successive inhibition (the "repressilator") can lead to a qualitative switch in system behavior among oscillatory and equilibrium dynamics. In all cases, the qualitative change in expression is due to the nonlinear nature of transcriptional feedback in which duplicated motifs interact via common pools of transcription factors. We are able to implicitly determine the critical values of copy number which lead to qualitative shifts in system behavior. In some cases, we are able to solve for the sufficient condition for the existence of a bifurcation in terms of kinetic rates of transcription, translation, binding, and degradation. We discuss the relevance of our findings to ongoing efforts to link copy number variation with cell fate determination by viruses, dynamics of synthetic gene circuits, and constraints on evolutionary adaptation.