Early work by Muller and McClintock discovered that the physical ends of linear chromosomes, named telomeres, possess an inherent ability to escape unwarranted fusions. Since then, extensive research has shown that this special feature relies on specialized proteins and structural properties that confer identity to the chromosome ends, thus allowing cells to distinguish them from intrachromosomal DNA double-strand breaks. Due to the inability of conventional DNA replication to fully replicate the chromosome ends and the downregulation of telomerase in most somatic human tissues, telomeres shorten as cells divide and lose this protective capacity. Telomere attrition causes the activation of the DNA damage checkpoint that leads to a cell-cycle arrest and the entering of cells into a nondividing state, called replicative senescence, that acts as a barrier against tumorigenesis. However, downregulation of the checkpoint overcomes this barrier and leads to further genomic instability that, if coupled with re-stabilization of telomeres, can drive tumorigenesis. This review focuses on the key experiments that have been performed in the model organism Saccharomyces cerevisiae to uncover the mechanisms that protect the chromosome ends from eliciting a DNA damage response, the conservation of these pathways in mammals, as well as the consequences of their loss in human cancer.