The ocular motor system provides several advantages for studying the brain, including well-defined populations of neurons that contribute to specific eye movements. Generation of rapid eye movements (saccades) depends on excitatory burst neurons (EBNs) and omnipause neurons (OPNs) within the brain stem; both types of cell are highly active. Experimental lesions of EBNs and OPNs cause slowing or complete loss of saccades. We report a patient who developed a permanent, selective saccadic palsy following cardiac surgery. When she died several years later, surprisingly, autopsy showed preservation of EBNs and OPNs. We therefore considered other mechanisms that could explain her saccadic palsy. Recent work has shown that both EBNs and OPNs are ensheathed by perineuronal nets (PNs), which are specialized extracellular matrix structures that may help stabilize synaptic contacts, promote local ion homeostasis, or play a protective role in certain highly active neurons. Here, we review the possibility that damage to PNs, rather than to the neurons they support, could lead to neuronal dysfunction—such as saccadic palsy. We also suggest how future studies could test this hypothesis, which may provide insights into the vulnerability of other active neurons in the nervous system that depend on PNs.