The spontaneous generation of inertia-gravity waves by balanced motion, and the consequent non-existence of an exactly invariant slow manifold, lead to fundamental limitations in the accuracy of balanced models. In the standard quasi-geostrophic regime (with small Rossby number and order-one rotational Froude number) the amplitude of the inertia-gravity waves can be expected to be exponentially small in the Rossby number. We demonstrate this explicitly by deriving asymptotic estimates for this amplitude in two models: the five-component model of Lorenz and Krishnamurthy; and a model describing the evolution of sheared disturbances in a three-dimensional Boussinesq fluid. In both cases the asymptotic estimates are confirmed by numerical experiments. We attribute the wave generation to a Stokes phenomenon, which is associated with the presence of singularities in the balanced motion for complex values of the time. Correspondingly, the details of the wave generation are well described by Berry's theory of the Stokes phenomenon. The concepts used our analysis (optimal truncation of asymptotic series, Borel-Laplace transform, complex-time dynamics) help clarify some of the issues still open in balanced dynamics and initialization. The relevance of our results to more complex,e.g. turbulent, systems is discussed.