Genetic variants that disrupt protein-coding DNA are ubiquitous in the human population, with ~100 such loss-of-function variants per individual. While most loss-of-function variants are rare, a subset have risen to high frequency and occur in a homozygous state in healthy individuals. It is unknown why these common variants are well-tolerated, even though some affect essential genes implicated in Mendelian disease. Here, we combine genomic, proteomic, and biochemical data to demonstrate that many common nonsense variants do not ablate protein production from their host genes. We provide direct evidence for previously proposed mechanisms of gene rescue such as alternative splicing and C-terminal truncation. Furthermore, we identify novel mechanisms of rescue, including alternative translation initiation at non-canonical start codons and stop codon readthrough. Our results suggest a molecular explanation for the mild fitness costs of common nonsense variants, and indicate that translational plasticity plays a prominent role in shaping human genetic diversity.