It is probably a little of both. Think of it this way (waaaaaaay oversimplified): humans can live a lot longer than we do. And we are living longer than we have.
Human biology allows us to live longer, but we do so as aided by technology and science: medicines and vitamins and exercise, etc.
Time Lords probably have a capability within their cells to regenerate to varrying degrees. Time Lord technology and evolution, maybe even psychological training allows them to do so.
We've seen lots of people from Gallifrey not regenerate. That dude who was a reporter in "The Deadly Assasain" for example. Jenny, the Doctor's daughter, was assumed by the Doctor not to be capable of regeneration, most likely due to her youth, and possibly even ignorance about what her body can do.
On serveral occations the Doctor has needed the help of the TARDIS to regenerate, and to survive the regeneration. The TARDIS "helped", but it was not 100% presponsible.