Examples are given of nonlinear, time-invariant systems in continuous-time, discrete-time, and of hybrid type, that have linear sector growth, the origin globally exponentially stable, and that can be driven to infinity by arbitrarily small additive decaying exponentials. Resulting observations about additive cascades and Lyapunov functions are discussed. These observations extend those derived from an example, which recently appeared in the literature, of a globally asymptotically stable continuous-time system destabilized by a piecewise constant, integrable disturbance.