Telomere dysfunction is typically studied under conditions in which a component of the six-subunit shelterin complex that protects chromosome ends is disrupted. The nature of spontaneous telomere dysfunction is less well understood. Here we report that immortalized human cell lines lacking wild-type p53 function spontaneously show many telomeres with a DNA damage response (DDR), commonly affecting only one sister chromatid and not associated with increased chromosome end-joining. DDR(+) telomeres represent an intermediate configuration between the fully capped and uncapped (fusogenic) states. In telomerase activity-positive (TA(+)) cells, DDR is associated with low TA and short telomeres. In cells using the alternative lengthening of telomeres mechanism (ALT(+)), DDR is partly independent of telomere length, mostly affects leading strand-replicated telomeres, and can be partly suppressed by TRF2 overexpression. In ALT(+) (but not TA(+)) cells, DDR(+) telomeres preferentially associate with large foci of extrachromosomal telomeric DNA and recombination proteins. DDR(+) telomeres therefore arise through different mechanisms in TA(+) and ALT(+) cells and have different consequences.