To appear in Statistical Mechanics Research Focus, Special volume (Nova Science Pub., NY, in press) (LaTex, 16 pgs, 14 figures).We investigate the effects of a time-correlated noise on an extended chaotic system. The chosen model is the Lorenz'96, a kind of toy model used for climate studies. The system is subjected to both temporal and spatiotemporal perturbations. Through the analysis of the system's time evolution and its time correlations, we have obtained numerical evidence for two stochastic resonance-like behaviors. Such behavior is seen when a generalized signal-to-noise ratio function are depicted as a function of the external noise intensity or as function of the system size. The underlying mechanism seems to be associated to a noise-induced chaos reduction. The possible relevance of those findings for an optimal climate prediction are discussed, using an analysis of the noise effects on the evolution of finite perturbations and errors.We acknowledge financial support from MEC, Spain, through Grant No. CGL2004-02652/CLI. JAR thanks the MEC, Spain, for the award of a Juan de la Cierva fellowship. HSW thanks the European Commission for the award of a Marie Curie Chair during part of the development of this work.Peer reviewed