This paper considers the observer problem for a class of nonlinear systems which are not observable, but can have a finite number of (non-converging) indistinguishable trajectories. This means that in general it does not exist a usual, i.e. single-valued, observer. Instead, a multivalued one is proposed here, allowing to estimate the full set of indistinguishable trajectories. By extending the system's dynamics, it is also possible to estimate all possible unknown inputs acting on the system. The approach is illustrated with various examples, including experiments with a magnetic levitation system.