No, you need to have something left for the skill to have any hope of activation.
In Niil's case, at wounded twice, when he tries to use Accelerate Healing, his effective skill is 0D+1, less than the 5 he would need.
There are ways to boost your effective skill though - spending a round or two in preparation gives bonus dice, using Control Pain can remove the effects of injury, Concentrate gives you bonus dice, you can spend Character Points, use Force Points, etc.