Abstract: Many methods for inferring phylogenetic trees and networks leverage quartets, offering both computational efficiency and a strong theoretical foundation under the multispecies coalescent model. However, the theoretical guarantees of these methods often rely on the assumption that the underlying evolutionary history fits a specific model, such as a tree in tree-based approaches or a level 1 network (that is, one without interlocking cycles) in many network-based methods. In this talk, we explore the challenges that arise when the true evolutionary process is more complex than the model assumes. For instance, we consider how tree inference methods perform on data generated from a network, or how network methods behave when the true network is not level-1.