We read that ageing is related to cell death when we run out of telomeres at the end of our DNA molecules. Humans live roughly for 70 years - the traditional three-score years and ten. This compares with the great apes, such as chimps and gorillas, which live for about 40 years. So we would expect the great apes to have fewer telomeres than humans. In fact the reverse is the case. Humans have telomeres of about 10 kilobases in length whereas the equivalent length in chimps and other great apes is about 23 kilobases. So what is going on? Why do we live longer?
The answer should be obvious: Telomeres do not determine an organism's longevity. If "we read" something that is trivially contradicted by observation, then, of course what "we" have read is wrong. In fact, if you actually read the primary literature on aging, telomeres play a fairly minor role. Unfortunately some sections of the media, and probably some educators who get their information from dumbed-down sensationalized media stories, have grabbed on to telomeres as a general explanation, without actually looking at the evidence.
There is not one simple cause of aging, like short telomeres, and the fact is that it hasn't even been settled whether aging is caused by evolution failing to evolve good enough repair mechanisms, or whether it evolved with a purpose, it might be that the genes of an organism are better spread if that organism doesn't live too long for example. For more details on the latter debate see this previous issue of Current Aging Science (all the articles are free).