There are many aspects to this topic, and the real answer is pretty complex. Cuttings from many types of plants get reinvigorated when they are removed from the parent plant and propagated. Some types of plants will even possibly revert to a juvenile form. There are some that will almost always revert to a juvenile form, some that almost never will, and some that are dependent upon various factors. When the plant material is reinvigorated, and possibly rejuvenated, it's life span is typically increased. Some vegetatively reproduced cultivars are many centuries old.

Although cuttings may seem to have their age "reset", it's not a perfect reset. Genetic mutation (often small and unnoticed on a short-term time scale) and acquired pathogens are some factors that would seem to limit how long a plant could be kept alive through vegetative propagation. I was planning on going into the mutation topic a little further, but am finding it overly complex to cover with a reasonable amount of typing. Maybe I will write more later or someone else will cover that more.

Well, to save you or someone else from typing out a chapter from a University-level book of propagation and mutations, I will just take the basic answer of, "It depends on the species of plant" and run with it. Though I'm sure it would be an interesting read. If I ever get to the point where I'm actually trying to develop certain mutations, I will have to go look for some books on the subject, but for now, for my current state of experience and knowledge in this area, the short answer works quite well. Still... you should know that you have piqued the interest of my geeky side and I will probably end up looking up more information on the subject before the end of the day.