Copy number variants affect both disease and normal phenotypic variation, but those lying within heavily duplicated, highly identical sequence have been difficult to assay. By analyzing short-read mapping depth for 159 human genomes, we demonstrated accurate estimation of absolute copy number for duplications as small as 1.9 kilobase pairs, ranging from 0 to 48 copies. We identified 4.1 million "singly unique nucleotide" positions informative in distinguishing specific copies and used them to genotype the copy and content of specific paralogs within highly duplicated gene families. These data identify human-specific expansions in genes associated with brain development, reveal extensive population genetic diversity, and detect signatures consistent with gene conversion in the human species. Our approach makes ~1000 genes accessible to genetic studies of disease association.