Telomeres, repetitive sequences located at the ends of most eukaryotic chromosomes, provide a mechanism to replenish terminal sequences lost during DNA replication, limit nucleolytic resection, and protect chromosome ends from engaging in double-strand break (DSB) repair. The ribonucleoprotein telomerase contains an RNA subunit that serves as the template for the synthesis of telomeric DNA. While telomere elongation is typically primed by a 3′ overhang at existing chromosome ends, telomerase can act upon internal non-telomeric sequences. Such de novo telomere addition can be programmed (for example, during chromosome fragmentation in ciliated protozoa) or can occur spontaneously in response to a chromosome break. Telomerase action at a DSB can interfere with conservative mechanisms of DNA repair and results in loss of distal sequences but may prevent additional nucleolytic resection and/or chromosome rearrangement through formation of a functional telomere (termed “chromosome healing”). Here, we review studies of spontaneous and induced DSBs in the yeast Saccharomyces cerevisiae that shed light on mechanisms that negatively regulate de novo telomere addition, in particular how the cell prevents telomerase action at DSBs while facilitating elongation of critically short telomeres. Much of our understanding comes from the use of perfect artificial telomeric tracts to “seed” de novo telomere addition. However, endogenous sequences that are enriched in thymine and guanine nucleotides on one strand (TG-rich) but do not perfectly match the telomere consensus sequence can also stimulate unusually high frequencies of telomere formation following a DSB. These observations suggest that some internal sites may fully or partially escape mechanisms that normally negatively regulate de novo telomere addition.