The fact that most cells in an organism's body can not divide indefinitely is undermine by a process called replicative senescence.
Few exceptions to this fact can be tumor cells as they completely loose the property of contact inhibition and stem cells.
This can be explained by the ‘telomere shortening theory’ that conveys that due to repeated cell division, the length of telomeres, that are present at the end of each chromosomes, keeps on shortening. After a definite number of divisions, the telomere disappears and thus the cells do not enter further into cell cycle.
Replicative senescence also indirectly gives us a reason for aging.