We consider a nonequilibrium, bistable system exposed to external, additive, Gaussian, colored noise. A new approximate theory, which results in a nonlinear Fokker-Planck type equation capable of accurately modeling the long time dynamics is constructed, and the mean sojourn time (MST) (in a potential well) is evaluated in the limits of weak noise and small correlation time. An electronic circuit which accurately mimics the bistable system has been constructed. Experimental measurements on this circuit are in good quantitative agreement with the predictions of the theory, which indicate an exponential increase of the MST with increasing noise correlation time. This observed exponential dependence is the clear signature that an Arrhenius law governs the process of escape from the well.