Fault detection and diagnosis is critical for maintaining the health of process systems. Common fault signals include process and disturbance parameter changes, as well as sensor and actuator malfunctions typically manifested as persistent drifts or abrupt biases. These may be characterized by the existence of latent "fault" states. This work examines the effectiveness of a hidden Markov model framework for modeling such fault regimes. The proposed methodology may be interpreted as a generalization of the commonly employed mixture-of-Gaussians approach and is demonstrated through a shell-and-tube heat exchanger problem. Furthermore, the flexibility of the method is shown in the context of detecting valve stiction, a significant problem in the process industries.