Chromosomal rearrangements, despite being detrimental to normal organismal fitness, are ubiquitous in cancer and often act as driver events. Copy number variations (CNVs) in the autosomal chromosomes are not known to have specific balancing mechanists but cancer cells can thrive with large dosage imbalances. Since the effect of CNVs on the cellular proteome of tumours is poorly understood, we analyzed recently generated proteogenomic data-sets on 282 patient samples of solid tumours, and investigated the implication of CNVs in the proteome of these cells. We found that the impact of CNVs is post-transcriptionally attenuated in 23-33% of the proteins with a strong enrichment for protein complex subunits. Interaction partners have a high co-regulation of their abundances and some complex subunits act as rate-limiting steps of complex assembly, indirectly controlling the abundance of other complex members. We identified 48 such regulatory interactions providing insight into assembly pathways of protein complexes involved in relevant biological processes in cancer. Lastly, we found that a gene-signature of the proteome attenuation is associated with increased resistance to chaperone and proteasome inhibitors. This study presents novel insights into the widespread importance of post-transcriptional mechanisms in cancer that allow cells to cope with their altered genomic profile.