We consider measurements from possibly zero-mean stochastic processes in a nonlinear filtering framework. This is a challenging problem, since it is only the second order properties of the measurements that bear information about the unknown state vector. The covariance function of the measurements can have both spatial and temporal correlation that depend on the state. Recently, a solution to this problem was presented for the case of Gaussian processes. We here extend the theory to Student's t processes. We illustrate the state observability by a simple but still realistic simulation example.