Recent data show that cells from many cancers exhibit massive chromosome instability. The traditional view is that the gradual accumulation of mutations in genes involved in transcriptional regulation and cell cycle controls results in tumor development. This, however, does not exclude the possibility that some mutations could be more potent than others in destabilizing the genome by targeting both chromosomal integrity and corresponding checkpoint mechanisms simultaneously. Three such examples of "single-hit" lesions potentially leading to heritable genome destabilization are discussed. They include: failure to release sister chromatid cohesion due to the incomplete proteolytic cleavage of cohesin; massive merotelic kinetochore misattachments upon condensin depletion; and chromosome under-replication. In all three cases, cells fail to detect potential chromosomal bridges before anaphase entry, indicating that there is a basic cell cycle requirement to maintain a degree of sister chromatid bridging that is not recognizable as chromosomal damage.