Most of the evidence does seem to infer that it's an ability that is granted (on graduation from the Academy?), rather than a natural thing. Maybe most Gallifreyans can't regenerate, but only Time Lords can?
Perhaps we can futher hypothosize that the '13 lives' thing is a limit imposed not on a biological "at birth" level, but as a condition of one becoming a Time Lord. It's infused into the ability to regenerate, that the body simply can't sustain more than 12 regenerative cycles.
It's worth noting that when the First Doctor regenerates, he is eager to rush back to the TARDIS before it happens, and the second Doctor talks about his renewal as being 'part of the TARDIS'. Most of the Doctor's regenerations happen to take place inside the TARDIS. Maybe the TARDIS provides merely a sanctuary, or perhaps 'she' has got properties that help the regeneration along? The Zero Room was apparently designed for this very purpose, hence why one exists both on Gallifrey and there's a replica in every TARDIS (or at least it was until the Doctor's Zero Room got jettisoned anyway).