State estimators for physical processes often must address different challenges, including nonlinear dynamics, states subject to hard constraints (e.g., nonnegative concentrations), and local optima. In this article, we compare the performance of two such estimators: the extended Kalman filter (EKF) and moving-horizon estimation (MHE). We outline conditions that lead to the formation of multiple optima in the estimator for systems tending to a steady state and propose tests that determine when these conditions hold for chemical reaction networks. Several simulation examples demonstrate estimation failure in the EKF, even in the absence of plant-model mismatch. We then examine the role that constraints play in determining the performance on these examples of MHE employing local optimization and a "smoothing" update for the arrival cost. This implementation of MHE represents a feasible, on-line alternative to the EKF for industrial practitioners. In each example, the two estimators are given exactly the same information, namely, tuning parameters, model, and measurements; yet MHE consistently provides improved state estimation and greater robustness to both poor guesses of the initial state and tuning parameters in comparison to the EKF. The only price of this improvement is the greater computational expense required to solve the MHE optimization.