  In chaotic dynamical systems, an infinitesimal perturbation is exponentially
amplified at a time-rate given by the inverse of the maximum Lyapunov exponent
$\lambda$. In fully developed turbulence, $\lambda$ grows as a power of the
Reynolds number. This result could seem in contrast with phenomenological
arguments suggesting that, as a consequence of `physical' perturbations, the
predictability time is roughly given by the characteristic life-time of the
large scale structures, and hence independent of the Reynolds number. We show
that such a situation is present in generic systems with many degrees of
freedom, since the growth of a non-infinitesimal perturbation is determined by
cumulative effects of many different characteristic times and is unrelated to
the maximum Lyapunov exponent. Our results are illustrated in a chain of
coupled maps and in a shell model for the energy cascade in turbulence.
