<p>Recent technological advances have enabled measurement of the synaptic wiring diagram, or ‘connectome’, of large neural circuits or entire brains. However, the extent to which such data constrain models of neural dynamics and function is debated. In this study, we developed a theory of connectome-constrained neural networks in which a ‘student’ network is trained to reproduce the activity of a ground truth ‘teacher’, representing a neural system for which a connectome is available. Unlike standard paradigms with unconstrained connectivity, the two networks have the same synaptic weights but different biophysical parameters, reflecting uncertainty in neuronal and synaptic properties. We found that a connectome often does not substantially constrain the dynamics of recurrent networks, illustrating the difficulty of inferring function from connectivity alone. However, recordings from a small subset of neurons can remove this degeneracy, producing dynamics in the student that agree with the teacher. Our theory demonstrates that the solution spaces of connectome-constrained and unconstrained models are qualitatively different and determines when activity in such networks can be well predicted. It can also prioritize which neurons to record to most effectively inform such predictions.</p>