Telomeres were defined by their ability to cap chromosome ends. Proteins with high affinity for the structure at chromosome ends, binding the G-rich, 3' single-stranded overhang at telomeres include Pot1 in humans and fission yeast, TEBP in Oxytricha nova and Cdc13 in budding yeast. Cdc13 is considered essential for telomere capping because budding yeast that lack Cdc13 rapidly accumulate excessive single-stranded DNA (ssDNA) at telomeres, arrest cell division and die. Cdc13 has a separate, critical role in telomerase recruitment to telomeres. Here, we show that neither Cdc13 nor its partner Stn1 are necessary for telomere capping if nuclease activities that are active at uncapped telomeres are attenuated. Recombination-dependent and -independent mechanisms permit maintenance of chromosomes without Cdc13. Our results indicate that the structure of the eukaryotic telomere cap is remarkably flexible and that changes in the DNA damage response allow alternative strategies for telomere capping to evolve.