The natural ends of chromosomes resemble DNA double-strand breaks (DSBs) and telomeres are therefore necessary to prevent recognition by the DNA damage response. The enzyme telomerase can also generate new telomeres at DSBs, resulting in the loss of genetic information distal to the break. How cells deal with different DNA ends is therefore an important decision. One critical point of regulation is to limit telomerase activity at DSBs and this is primarily accomplished in budding yeast by the telomerase inhibitor Pif1. Here we use Pif1 as a sensor to gain insight into the cellular decision at DSB ends with increasing telomeric character. We uncover a striking transition point in which 34 bp of telomeric (TG1-3)n repeat sequence is sufficient to render a DNA end insensitive to Pif1, thereby facilitating extension by telomerase. This phenomenon is unlikely to be due to Pif1 modification and we propose that Cdc13 confers a unique property to the TG34 end that prevents Pif1 action. We identify novel Cdc13 mutations that resensitize DNA ends to Pif1 and discover that many Cdc13 telomerase-null mutations are dependent on Pif1 status. Finally, the observed threshold of Pif1 activity recapitulates several properties of both DSBs and telomeres and we propose that this is the dividing line between these entities.