During David Tennant's first appearance (Christmas Special), The Doctor was able to regenerate an amputated hand with leftover regeneration energy from an hours-earlier regeneration. Later, he took a full blast from a Dalek weapon -- which at that time would often completely disintegrate a human -- and regenerated without even changing his face.
Going back further, when Jon Pertwee regenerated into Tom Baker, he was effectively dead (from radiation, as I recall), and the regeneration hadn't started on its own, but required a little "push" from another Time Lord he'd encountered in the same story.
It seems at one point that UNIT was of the opinion that they could kill Missy fast enough to keep her from regenerating, by using seven snipers -- "two shots for each heart, two for the brain, and one extra just for good measure" is the quote I recall -- which would imply there is some level of injury that would prevent regeneration (though whether Missy agreed with their estimate was left up in the air). There was also the original version of the incident in which River Song shot Matt Smith, waited for regeneration to begin, and shot him again to prevent the regeneration (though in fact Matt Smith's Doctor didn't have any regenerations left at that time, anyway).
In summation, I'd say that while there doesn't appear to be a canon statement of how much injury might prevent regeneration, it appears there is some level that would. As always with Doctor Who, however, canon is kind of timey-wimey, wibbly-wobbly.