Cell proliferation limit and senescence of embryonic stem cells and fibroblasts
Telomeres (caps on the ends of chromosomes that are gradually shortened during each cell division) determine the maximum number of times a cell can divide, known as the 'Hayflick limit'. Cells that are able to express telomerase, an enzyme capable of extending the telomeres, can divide indefinitely.

There are various types of stem cell in a body which replenish the 'pool' of cells in the tissue once they reach the end of their useful lifespan. Each stem cell expresses telomerase to lengthen the telomeres after a round of cell division. The inhibition of telomerase in somatic (normal) cells is a major 'challenge' for cancerous cells - in order to achieve uninhibited proliferation, they must express telomerase. Thus, there are many molecular similarities between stem cells and cancerous cells (well, with regards to telomerase, anyway!) [ref].

Fibroblasts are somatic cells - they are the 'normal' cells that make up the vast majority of organsisms. The only cells not known as somatic are gametes/germ cells and stem cells. They thus do not express telomerase, and have a replicative lifespan, also known as a Hayflick limit.

With regard to your question about engineering organs, there are many resident stem cells in tissues (and thus organs) that 'top-up' the somatic cells when required. As organisms age, this pool of stem cells becomes depleted [ref]. Artificially engineering an organ such as the heart (I assume this is what you mean) would be inherently difficult, as many vascular cells are not replaced, but can last a whole lifetime - this is also true for many neuronal cells. So replicative lifespan is not necessarily related to lifespan!