We study the consistency of response of nonlinear dynamical systems driven by chaotic and colored noise signals. We use the Lorenz model and a microchip solid-state laser to test consistency. Parameters are chosen so the system is in a chaotic state and response of the system starting at different initial conditions is studied for repeated applications of the drive signal. The convergence of the response is analyzed in dynamical phase space in terms of the local conditional Lyapunov exponent. The dependence of the local conditional Lyapunov exponent on drive signal and the existence of multiple basins of consistency are investigated.