Food web theory states that a weak interactor which is positioned in the food web such that it tends to deflect, or mute, energy away from a potentially oscillating consumer-resource interaction often enhances community persistence and stability. Here we examine how adding other weak interactions (predation/harvesting) on the stabilizing weak interactor alters the stability of food web using a set of well-established food web models/modules. We show that such "weak on weak" interaction chains drive an indirect dynamic cascade that can rapidly ignite a distant consumer-resource oscillator. Nonetheless, we also show that the "weak on weak" interactions are still more stable than the food web without them, and so weak interactions still generally act to stabilize food webs. Rather, these results are best interpreted to say that the degree of the stabilizing effect of a given important weak interaction can be severely compromised by other weak interactions (including weak harvesting).