Qualitative analytic estimates of the stochasticity limit of one- and many-dimensional nonlinear oscillating systems are derived using the overlapping of first-order resonances as a criterion for stochasticity. Computational results obtained with several very simple transformations are compared with the analytic estimates. Numerical studies significantly below the stochasticity limit of a many-dimensional nonlinear system reveal an example of a very slow instability, the first, to the best of our knowledge.