Chromosomal rearrangements, despite being detrimental, are ubiquitous in cancer and often act as driver events. The effect of copy number variations (CNVs) on the cellular proteome of tumours is poorly understood. Therefore, we have analysed recently generated proteogenomic data-sets on 282 tumour samples to investigate the impact of CNVs in the proteome of these cells. We found that CNVs are post-transcriptionally attenuated in 23-33% of proteins with an enrichment for protein complexes. Complex subunits are highly co-regulated and some act as rate-limiting steps of complex assembly, indirectly controlling the abundance of other complex members. We identified 48 such regulatory interactions and experimentally validated AP3B1 and GTF2E2 as controlling subunits. Lastly, we found that a gene-signature of protein attenuation is associated with increased resistance to chaperone and proteasome inhibitors. This study highlights the importance of post-transcriptional mechanisms in cancer which allow cells to cope with their altered genomes.