Despite whole-genome sequencing (WGS), many cases of single-gene disorders remain unsolved, impeding diagnosis and preventative care for people whose disease-causing variants escape detection. Since early WGS data analytic steps prioritize protein-coding sequences, to simultaneously prioritize variants in non-coding regions rich in transcribed and critical regulatory sequences, we developed GROFFFY, an analytic tool that integrates coordinates for regions with experimental evidence of functionality. Applied to WGS data from solved and unsolved hereditary hemorrhagic telangiectasia (HHT) recruits to the 100,000 Genomes Project, GROFFFY-based filtration reduced the mean number of variants/DNA from 4,867,167 to 21,486, without deleting disease-causal variants. In three unsolved cases (two related), GROFFFY identified ultra-rare deletions within the 3' untranslated region (UTR) of the tumor suppressor SMAD4, where germline loss-of-function alleles cause combined HHT and colonic polyposis (MIM: 175050). Sited >5.4 kb distal to coding DNA, the deletions did not modify or generate microRNA binding sites, but instead disrupted the sequence context of the final cleavage and polyadenylation site necessary for protein production: By iFoldRNA, an AAUAAA-adjacent 16-nucleotide deletion brought the cleavage site into inaccessible neighboring secondary structures, while a 4-nucleotide deletion unfolded the downstream RNA polymerase II roadblock. SMAD4 RNA expression differed to control-derived RNA from resting and cycloheximide-stressed peripheral blood mononuclear cells. Patterns predicted the mutational site for an unrelated HHT/polyposis-affected individual, where a complex insertion was subsequently identified. In conclusion, we describe a functional rare variant type that impacts regulatory systems based on RNA polyadenylation. Extension of coding sequence-focused gene panels is required to capture these variants.