Childhood learning difficulties and developmental disorders are common, but progress toward understanding their underlying brain mechanisms has been slow. Structural neuroimaging, cognitive, and learning data were collected from 479 children (299 boys, ranging in age from 62 to 223 months), 337 of whom had been referred to the study on the basis of learning-related cognitive problems. Machine learning identified different cognitive profiles within the sample, and hold-out cross-validation showed that these profiles were significantly associated with children's learning ability. The same machine learning approach was applied to cortical morphology data to identify different brain profiles. Hold-out cross-validation demonstrated that these were significantly associated with children's cognitive profiles. Crucially, these mappings were not one-to-one. The same neural profile could be associated with different cognitive impairments across different children. One possibility is that the organization of some children's brains is less susceptible to local deficits. This was tested by using diffusion-weighted imaging (DWI) to construct whole-brain white-matter connectomes. A simulated attack on each child's connectome revealed that some brain networks were strongly organized around highly connected hubs. Children with these networks had only selective cognitive impairments or no cognitive impairments at all. By contrast, the same attacks had a significantly different impact on some children's networks, because their brain efficiency was less critically dependent on hubs. These children had the most widespread and severe cognitive impairments. On this basis, we propose a new framework in which the nature and mechanisms of brain-to-cognition relationships are moderated by the organizational context of the overall network.