We consider the problem of estimating the state of a discrete-time, linear stochastic system whose observation process consists of a finite set of known, linear measurement models. The correspondence of the measurements with the models is unknown. Assuming all the measurements relate to the state, we derive a recursive algorithm, the multiple simultaneous measurement filter (MSMF), which provides a fixed-complexity, sub-optimal solution to this problem. The MSMF is a generalization of the discrete-time Kalman filter and reduces to the latter when a single measurement model applies. Simulations are provided that demonstrate the superior performance of the MSMF over the probabilistic data association filter for tracking a single target in an environment where up to three measurements of the state are received simultaneously.