Mathematical models can aid the design of genetic circuits, but may yield inaccurate results if individual parts are not modeled at the appropriate resolution. To illustrate the importance of this concept, we study transcriptional cascades consisting of two inducible synthetic transcription factors connected in series. Despite the simplicity of this design, we find that accurate prediction of circuit behavior requires mapping the dose responses of each circuit component along the dimensions of both its expression level and its inducer concentration. Using this multidimensional characterization, we were able to computationally explore the behavior of 16 different circuit designs. We experimentally verified a subset of these predictions and found substantial agreement. This method of biological part characterization enables the use of models to identify (un)desired circuit behaviors prior to experimental implementation, thus shortening the design-build-test cycle for more complex circuits.