It is not typically the bent of infrastructure to be continually responsive in a way that is expansive and inclusive; instead, for newcomers or those with alternative histories, aims, vision, values, and perspectives, the inertia of infrastructure is more likely to be experienced as infrastructural breakdowns. We ask: What might wisdom look like in these “structured” encounters? That is, what is the intellectual work of rhetoric on those thin ledges where institutional chronos shapes and limits possibilities for knowledge work and working relationships among people who likely would not have otherwise met? In response, we advance a framework for a constructive approach to infrastructure—one that prizes deliberation over rationalization and actively attends to the warrants underlying calls for public engagement. We first consider the relationship between infrastructure, rhetorical wisdom, and the imagination of possibilities, then lay out a framework for cultivating rhetorical wisdom in response to infrastructure breakdowns.