Telomeres were first recognized as a bona fide constituent of the chromosome based on their inability to rejoin with broken chromosome ends produced by radiation. Today, we recognize two essential and interrelated properties of telomeres. They circumvent the so-called end-replication problem faced by genomes composed of linear chromosomes, which erode from their termini with each successive cell division. Equally vital is the end-capping function that telomeres provide, which is necessary to deter chromosome ends from illicit recombination. This latter property is critical in facilitating the distinction between the naturally occurring DNA double-strand breaks (DSBs) found at chromosome ends (i.e., telomeres) and DSBs produced by exogenous agents. Here we discuss, in a brief historical narrative, key discoveries that led investigators to appreciate the unique properties of telomeres in protecting chromosome ends, and the consequences of telomere dysfunction, particularly as related to recombination involving radiation-induced DSBs.