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.
There's only a vague correlation between telomere length and a species' lifespan. See Comparative biology of mammalian telomeres: hypotheses on ancestral states and the roles of telomeres in longevity determination for a starting point. Telomeres are a part of the longevity story, but probably a fairly small part.
The Hayflick limit is generally associated with telomere length. Human telomeres are a little on the long side as species go, but are not extraordinary. Many species of mice, and other rodents, have far longer telomeres than humans, for example, and obviously have much shorter lifespans. There's also such a thing as a "mega-telomere", found in a number of bird species, which can hundreds of times longer than human telomeres.
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).