Posterior Cramér-Rao bounds (CRBs) are derived for the estimation performance of three Gaussian process-based state-space models. The parametric CRB is derived for the case with a parametric state transition and a Gaussian process-based measurement model. We illustrate the theory with a target tracking example and derive both parametric and posterior filtering CRBs for this specific application. Finally, the theory is illustrated with a positioning problem, with experimental data from an office environment where the obtained estimation performance is compared to the derived CRBs.