We consider finite model approximations of a large class of static and dynamic team problems where these models are constructed through uniform quantization of the observation and action spaces of the agents. The strategies obtained from these finite models are shown to approximate the optimal cost with arbitrary precision under mild technical assumptions. In particular, quantized team policies are asymptotically optimal. This result is then applied to Witsenhausen's celebrated counterexample and the Gaussian relay channel problem. For Witsenhausen's counterexample, our approximation approach provides, to our knowledge, the first rigorously established result that one can construct an ε-optimal strategy for any ε > 0 through a solution of a simpler problem.