In fact, the possibility is so great that they are currently trying to bring back the Mammoth (or the Mastodon, I'm not sure which) by trying to get non-degraded cells (sperm would be the best) and impregnate a modern elephant with it through artificial insemination.
Now granted that the offspring would be a half breed, but with careful breeding and re-introduction of the original creature's DNA, over a few generations you would have a creature that is 90%+ the original animal.
In a sense, bringing back an extinct species.
They are currently looking through all specimens of preserved thylacine (Tasmanian Tiger) in the hope of finding enough intact DNA in an attempt to bring them back.
Assuming that they're not already back. There have been numerous sightings...all unconfirmed.
Could we bring back a T-Rex? Likely not. At least not with the tech we have today.