Adaptive systems which are based on a posteriori parameter update algorithms are shown to possess a specific type of ’self-stabilization’ property, termed burst recovery. This results in bounded-disturbance bounded-error stability, provided that unbounded parameter drift does not occur. Persistent excitation prevents unbounded parameter drift, but only when both upper and lower bounds on excitation exist. For many adaptive systems, an upper bound on excitation cannot be assured using standard perturbation analyses, which require the disturbance to be ’suitably small’ compared to the lower bound on excitation. This paper proves that burst recovery in conjunction with only a lower bound on excitation provides bounded-disturbance bounded-error stability. The disturbance need not be small, and additional algorithm modifications (such as parameter projection) are not necessary.