A prevalent but untested paradigm is often used to describe the prominence-cavity system: the cavity is under-dense because it is evacuated by supplying mass to the condensed prominence. The thermal non-equilibrium (TNE) model of prominence formation offers a theoretical framework to predict the thermodynamic evolution of the prominence and the surrounding corona. We examine the evidence for a prominence-cavity connection by comparing the TNE model with diagnostics of dynamic extreme ultraviolet (EUV) emission surrounding the prominence, specifically prominence horns. Horns are correlated extensions of prominence plasma and coronal plasma which appear to connect the prominence and cavity. The TNE model predicts that large-scale brightenings will occur in the Solar Dynamics Observatory Atmospheric Imaging Assembly 171 Å bandpass near the prominence that are associated with the cooling phase of condensation formation. In our simulations, variations in the magnitude of footpoint heating lead to variations in the duration, spatial scale, and temporal offset between emission enhancements in the other EUV bandpasses. While these predictions match well a subset of the horn observations, the range of variations in the observed structures is not captured by the model. We discuss the implications of our one-dimensional loop simulations for the three-dimensional time-averaged equilibrium in the prominence and the cavity. Evidence suggests that horns are likely caused by condensing prominence plasma, but the larger question of whether this process produces a density-depleted cavity requires a more tightly constrained model of heating and better knowledge of the associated magnetic structure.