Copy-number variants (CNVs) that increase the risk for neurodevelopmental disorders also affect cognitive ability. However, such CNVs remain challenging to study due to their scarcity, limiting our understanding of gene-dosage-sensitive biological processes linked to cognitive ability. We performed a genome-wide association study (GWAS) in 258,292 individuals, which identified—for the first time—a duplication at 2q12.3 associated with higher cognitive performance. We developed a functional-burden analysis, which tested the association between cognition and CNVs disrupting 6,502 gene sets biologically defined across tissues, cell types, and ontologies. Among those, 864 gene sets were associated with cognition, and effect sizes of deletion and duplication were negatively correlated. The latter suggested that functions across all biological processes were sensitive to either deletions (e.g., subcortical regions, postsynaptic) or duplications (e.g., cerebral cortex, presynaptic). Associations between non-brain tissues and cognition were driven partly by constrained genes, which may shed light on medical comorbidities in neurodevelopmental disorders.