How much does the Hayflick limit/telomere length vary across taxa and within humans?
Many studies, including those of Hayflick himself, have found a strong correlation between a species' Hayflick limit and its maximum lifespan. For example, Galapagos tortoises, which can live over a century, have a Hayflick limit of 90-125, and mice, which live only a few years, have a Hayflick limit of 14-28.

However, the mechanism connecting the maximum number of cell divisions and maximum lifespan is not entirely clear. Cell division is associated much more with growth and development than with the maintenance of mature tissues. Cells that have stopped dividing can be perfectly functional for decades. Cells that don't stop dividing are called "cancer". So, while the correlation is strong, the reason for the correlation remains something of a mystery.

Rose, Michael R. 2005. The Long Tomorrow: How Advances in Evolutionary Biology Can Help us Postpone Aging. Oxford University Press, New York. pp 20-23