Researchers in some of the most active fields of science, including, e.g., geophysics or systems biology, have to deal with very-large-scale stochastic dynamic models of real world phenomena for which conventional prediction and estimation methods are not well suited. In this paper, we investigate the application of a novel nested particle filtering scheme for joint Bayesian parameter estimation and tracking of the dynamic variables in a high dimensional state space model-namely a stochastic version of the two-scale Lorenz 96 chaotic system, commonly used as a benchmark model in meteorology and climate science. We provide theoretical guarantees on the algorithm performance, including uniform convergence rates for the approximation of posterior probability density functions of the fixed model parameters.