Structural variants (SVs) comprise the largest genetic variants, altering from 50 base pairs to megabases of DNA. However, SVs have not been effectively ascertained in most genetic association studies, leaving a key gap in our understanding of human complex trait genetics. We ascertained protein-altering SVs from UK Biobank whole-exome sequencing data (n=468,570) using haplotype-informed methods capable of detecting sub-exonic SVs and variation within segmental duplications. Incorporating SVs into analyses of rare variants predicted to cause gene loss-of-function (pLoF) identified 100 associations of pLoF variants with 41 quantitative traits. A low-frequency partial deletion of RGL3 exon 6 appeared to confer one of the strongest protective effects of gene LoF on hypertension risk (OR = 0.86 [0.82-0.90]). Protein-coding variation in rapidly-evolving gene families within segmental duplications-previously invisible to most analysis methods-appeared to generate some of the human genome's largest contributions to variation in type 2 diabetes risk, chronotype, and blood cell traits. These results illustrate the potential for new genetic insights from genomic variation that has escaped large-scale analysis to date.