Understanding how species extinctions affect communities of interacting species is an important challenge of ecology. The presence of unfeasible interactions, termed forbidden links, due to physiological or morphological barriers, is likely to decrease the plasticity of interaction networks, affecting their robustness to species extinctions. However, the existence of these forbidden links has been long debated, stressing the need to learn more about their prevalence and putative consequences for community robustness to extinctions. To tackle this gap, we used a dataset of plant-hummingbird interactions collected in Brazil, Costa Rica and Ecuador along elevation gradients. We used a Bayesian hierarchical model, to assess the probabilistic importance of exploitation barriers in determining species interactions. We found evidence for exploitation barriers between flowers and hummingbirds with a shorter bill than the corolla. We also showed that the proportion of forbidden links can change drastically among communities, because of changes in trait distributions. Finally we showed that networks with a high proportion of forbidden links had low robustness, because of constraints on interaction rewiring. Our results suggest that exploitation barriers are not rare and strongly limit interaction rewiring, and thus the rescue of species experiencing partner extinction may be limited.