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Thin, flexible sheets can be patterned and bonded to form internal fluidic networks, which enable actuation, sensing, and control, but failure of these sheet-based systems—and how to take advantage of this failure—remains relatively unexplored. Here, we examine this concept using heat-sealable textiles as a material platform. We determine the effects of geometry and material processing on bond strength and burst pressure; these findings can ensure a sheet-based fluidic system is sufficiently robust for a given use case. Building on this framework, we introduce a fuse-like component into which failure is deliberately programmed. In addition to limiting damage in the case of overpressurization, we leverage this programmed failure to enable distinct capabilities including (1) the binary selection of operating modes and (2) the sequencing of a series of tasks with a single pressure input. These findings will facilitate the development of more intelligent sheet-based fluidic systems.