Abstract Background Copy number variants (CNVs), including deletions, amplifications, and other rearrangements, are common in human and cancer genomes. Copy number data from array comparative genome hybridization (aCGH) and next-generation DNA sequencing is widely used to measure copy number variants. Comparison of copy number data from multiple individuals reveals recurrent variants. Typically, the interior of a recurrent CNV is examined for genes or other loci associated with a phenotype. However, in some cases, such as gene truncations and fusion genes, the target of variant lies at the boundary of the variant. Results We introduce Neighborhood Breakpoint Conservation (NBC), an algorithm for identifying rearrangement breakpoints that are highly conserved at the same locus in multiple individuals. NBC detects recurrent breakpoints at varying levels of resolution, including breakpoints whose location is exactly conserved and breakpoints whose location varies within a gene. NBC also identifies pairs of recurrent breakpoints such as those that result from fusion genes. We apply NBC to aCGH data from 36 primary prostate tumors and identify 12 novel rearrangements, one of which is the well-known TMPRSS2-ERG fusion gene. We also apply NBC to 227 glioblastoma tumors and predict 93 novel rearrangements which we further classify as gene truncations, germline structural variants, and fusion genes. A number of these variants involve the protein phosphatase PTPN12 suggesting that deregulation of PTPN12, via a variety of rearrangements, is common in glioblastoma. Conclusions We demonstrate that NBC is useful for detection of recurrent breakpoints resulting from copy number variants or other structural variants, and in particular identifies recurrent breakpoints that result in gene truncations or fusion genes. Software is available at http://http.//cs.brown.edu/people/braphael/software.html.