Applying the Kalman filtering scheme to linearized system dynamics and observation models does in general not yield optimal state estimates. More precisely, inconsistent state estimates and covariance matrices are caused by neglected linearization errors. This paper introduces a concept for systematically predicting and updating bounds for the linearization errors within the Kalman filtering framework. To achieve this, an uncertain quantity is not characterized by a single probability density anymore, but rather by a set of densities and accordingly, the linear estimation framework is generalized in order to process sets of probability densities. By means of this generalization, the Kalman filter may then not only be applied to stochastic quantities, but also to unknown but bounded quantities. In order to improve the reliability of Kalman filtering results, the last-mentioned quantities are utilized to bound the typically neglected nonlinear parts of a linearized mapping.