The 2 degree-of-freedom elastic pendulum equations can be considered as the lowest order analogue of interacting low-frequency (slow) Rossby and high-frequency (fast) gravity waves in the atmosphere. The strength of the coupling between the low and the high frequency waves is controlled by a single coupling parameter, defined by the ratio of the fast and slow characteristic time scales. Efficient, high accuracy, and symplectic structure preserving numerical solutions are designed for the elastic pendulum equation in order to study the role balanced dynamics play in local predictability. To quantify changes in the local predictability, two measures are considered: the local Lyapunov number and the leading singular value of the tangent linear map. A 3-dimensional variational data assimilation scheme for the elastic pendulum is also developed. To obtain an estimate of the typical size of the uncertainty in the initial conditions, Observing System Simulation Experiments (OSSE) are carried out by using the new data assimilation system. It is shown, both based on theoretical considerations and numerical experiments, that there exist regions of the phase space where the local Lyapunov number indicates exceptionally high predictability, while the dominant singular value indicates exceptionally low predictability. It is also demonstrated that, when the model is in a nearly balanced state, the local Lyapunov number has a tendency to choose instabilities associated with balanced motions, while the dominant singular value favors instabilities related to highly unbalanced motions. It is demonstrated that there is a statistically highly significant linear relationship between the amplification of the infinitesimal and the finite size uncertainties (representative for the analysis uncertainty). In the presence of finite size initial uncertainties, the local Lyapunov number, which has been derived by assuming infinitesimal initial perturbations, provides an increasingly more reliable estimate of the error growth as predictability decreases. The implications of these findings for atmospheric dynamics and numerical weather prediction are discussed.