The cloned sheep, Dolly, was said to have died very soon because the cells used to create it were taken from an adult sheep with an aged telomere. Why doesn't this happen with humans? Why aren't we born with 25 years less lifespan?
$\begingroup$ Our telomeres shorten as well with every division, except for stem cells (which have an active telomerase). The maximum number of divisions is called Hayflick limit. $\endgroup$– Chris ♦Mar 4, 2015 at 17:01
Broadly speaking there are two classes of cell in an organism:
- These are by far the most numerous in your body, and are the differentiated cells including everything from retinal cells to liver cells to neurons.
Stem cells and germ cells
- Stem cells reside in tissues and replenish the somatic cells as they are damaged or removed, or as the organ grows,
- Germ cells - in particular oocytes in women's ovaries - are pools of immature eggs that are generated when a women is still a foetus in her own mothers womb!
The difference between these two types that relates to your question is that somatic cells have a limited replicative lifespan - a limit to the number of times they can divide - because their telomeres (caps to the ends of chromosomes) erode during cell division, whereas stem cells are able express the enzyme telomerase which extends the telomeres and allows cells limitless replicative potential (excluding other factors). Replenishing telomeres is a major step in cancer progression .
Embryonic cells have 'complete' telomeres when they are fertilised and begin to proliferate into a new organism. Interestingly the telomere length of the germ cells is not at all related to fertility , suggesting to me that telomerase is expressed for a while post-fertilisation, so the length prior to fertilisation is not important.
Dolly was produced by a process called Somatic Cell Nuclear Transferase, where the nucleus from a somatic cell from an adult organism is transplated into an embryo with its nucleus removed . A study found that she had consistently shorter telomeres corresponding to the fact that the original donor was 6-years old already 
You might also be interested in my answers to these questions
Cell proliferation limit and senescence of embryonic stem cells and fibroblasts
Difference In Telomeres Between A Thale Cress Plant And A Methuselah Tree
- Shay, 2001, Human Molecular Genetics. Telomerase and cancer. http://hmg.oxfordjournals.org/content/10/7/677.full
- Turner, 2013, Mol Hum reprod. Telomere lengths in human pronuclei, oocytes and spermatozoa. http://www.ncbi.nlm.nih.gov/pubmed/23519357
- Shiels, 1999. Cloning. Analysis of telomere length in Dolly, a sheep derived by nuclear transfer. http://www.ncbi.nlm.nih.gov/pubmed/16218837