Constrained state estimation in nonlinear/non-Gaussian processes has been the domain of optimization based methods such as moving horizon estimation (MHE). MHE has a Bayesian interpretation, but it is not practical to implement a recursive MHE without assumptions of Gaussianity and linearized dynamics at various stages. This paper presents the constrained cell filter (CCF) as an alternative to MHE, requiring no linearization, jacobians, or nonlinear program. The CCF computes a piecewise constant approximation of the state probability density function with support defined by constraints; thus, all point estimates are constrained. The CCF can be more accurate and orders of magnitude faster than MHE for problems of a size as investigated in this work.