One persistent challenge in scientific practice is that the structure of the world can be unstable: changes in the broader context can alter which model of a phenomenon is preferred, all without any overt signal. Scientific discovery becomes much harder when we have a moving target, and the resulting incorrect understandings of relationships in the world can have significant real-world and practical consequences. In this paper, we argue that it is common (in certain sciences) to have changes of context that lead to changes in the relationships under study, but that standard normative accounts of scientific inquiry have assumed away this problem. At the same time, we show that inference and discovery methods can "protect" themselves in various ways against this possibility by using methods with the novel methodological virtue of "diligence." Unfortunately, this desirable virtue provably is incompatible with other desirable methodological virtues that are central to reliable inquiry. No scientific method can provide every virtue that we might want.