Accurate and interpretable car-following models are essential for traffic simulation and autonomous vehicle development. However, classical models like the Intelligent Driver Model (IDM) are fundamentally limited by their parsimonious and single-regime structure. They fail to capture the multi-modal nature of human driving, where a single driving state (e.g., speed, relative speed, and gap) can elicit many different driver actions. This forces the model to average across distinct behaviors, reducing its fidelity and making its parameters difficult to interpret. To overcome this, we introduce a regime-switching framework that allows driving behavior to be governed by different IDM parameter sets, each corresponding to an interpretable behavioral mode. This design enables the model to dynamically switch between interpretable behavioral modes, rather than averaging across diverse driving contexts. We instantiate the framework using a Factorial Hidden Markov Model with IDM dynamics (FHMM-IDM), which explicitly separates intrinsic driving regimes (e.g., aggressive acceleration, steady-state following) from external traffic scenarios (e.g., free-flow, congestion, stop-and-go) through two independent latent Markov processes. Bayesian inference via Markov chain Monte Carlo (MCMC) is used to jointly estimate the regime-specific parameters, transition dynamics, and latent state trajectories. Experiments on the HighD dataset demonstrate that FHMM-IDM uncovers interpretable structure in human driving, effectively disentangling internal driver actions from contextual traffic conditions and revealing dynamic regime-switching patterns. This framework provides a tractable and principled solution to modeling context-dependent driving behavior under uncertainty, offering improvements in the fidelity of traffic simulations, the efficacy of safety analyses, and the development of more human-centric ADAS.