  According to the classical theory of Brownian motion, the mean squared
displacement of diffusing particles evolves linearly with time whereas the
distribution of their displacements is Gaussian. However, recent experiments on
mesoscopic particle systems have discovered Brownian yet non-Gaussian regimes
where diffusion coexists with an exponential tail in the distribution of
displacements. Here we show that, contrary to the present theoretical
understanding, the length scale $\lambda$ associated to this exponential
distribution does not necessarily scale in a diffusive way. Simulations of
Lennard- Jones systems reveal a behavior $\lambda \sim t^{1/3}$ in three
dimensions and $\lambda \sim t^{1/2}$ in two dimensions. We propose a scaling
theory based on the idea of hopping motion to explain this result. In contrast,
simulations of a tetrahedral gelling system, where particles interact by a
non-isotropic potential, yield a temperature-dependent scaling of $\lambda$. We
interpret this behavior in terms of an intermittent hopping motion. Our
findings link the Brownian yet non-Gaussian phenomenon with generic features of
glassy dynamics and open new experimental perspectives on the class of
molecular and supramolecular systems whose dynamics is ruled by rare events.
