  When a quantum nonlinear system is linearly coupled to an infinite bath of
harmonic oscillators, quantum coherence of the system is lost on a decoherence
time-scale $\tau_D$. Nevertheless, quantum effects for observables may still
survive environment-induced decoherence, and be observed for times much larger
than the decoherence time-scale. In particular, we show that the Ehrenfest
time, which characterizes a departure of quantum dynamics for observables from
the corresponding classical dynamics, can be observed for a quasi-classical
nonlinear oscillator for times $\tau \gg\tau_D$. We discuss this observation in
relation to recent experiments on quantum nonlinear systems in the
quasi-classical region of parameters.
