I'm happy to see that I'm not the only one that has the strange and sudden thoughts. I think it's a complicated answer. We would need to take all the answers above mine and do a mash-up of them to reach a definite conclusion. I agree on the dark magic wound subject. <PERSON> lost and never regrew his body parts because he probably fell to a curse each time. <PERSON> can't regrow his ear because it was dark magic that took it. So we can conclude that dark magic wounds can't be regrown.
<PERSON>'s dialogue in chapter 5 of the Deathly Hallows gives also an indication. She COULD'VE regrown his ear if it was not for dark magic. Now the ear is flesh and cartilage. It also contains nerves but not in a huge quantity. So we can assume that flesh, cartilage and nerves can be regrown to an extent.
Now it's the extent that needs to be discussed. The ear is a rather small body part but what would happen if a hand or a leg was cut? These are bigger and more complex body parts. We know wizarding doctors can regrow bones with the use of Skele-gro which is pretty impressive in itself. So for example if someone lost the lower half of his leg and he was given Skele-gro, would the bone grow back out in the open or does still having the flesh around but empty inside make a difference? What <PERSON> said about Professor <PERSON> is interesting. 722
Why didn't the Professor have them regrow (if it's possible)? Is it the circumstances? Was he in a position where he couldn't be treated immediately and then it was too late? Is it the type of wound? People assume that <PERSON> didn't lose his limbs by dark magic but what kind of non-dark magic creature would make you lose a limb in the first place? We have to consider that it might also be a joke or word play from <PERSON>.
There is also the medic that needs to be taken in consideration. While <PERSON> isn't bad, she isn't the best either. There might be lots of medics at St Mungo that are better than her and have more experience regarding our case. There might be experts on limb regrowning. And we haven't encountered them in the books (mainly because we thankfully didn't need them).
On the subject of splinching, I don't think it can be used as an example or a basis. The two examples given above kind of contradict themselves. After <PERSON>'s splinching, in the book and in the movie we know he had been badly injured. He lost some of his flesh as if someone had peeled some parts of his skin off to the muscle under it. In the movie there's seemingly no blood, probably to keep the rating down but it looked pretty bad. BUT he technically didn't lose a limb? After <PERSON>'s splinching, she apparently lost her leg. BUT with just a flick of her wand, <PERSON> or another head of house, it's not really clear who, set things straight. I'm sure a flick of a wand wouldn't be enough to reattach a hand some lost because it was cut off with a knife. And when <PERSON> "lost" her leg she didn't fall down from the pain, blood gushing from her stump so I think splinching in general is irrelevant and can't be taken in consideration.
Personally I think it should be possible. In <PERSON>'s universe so many things can be done with magic, why would regrowning a limb be impossible? I think hellfire17 gave an interesting and valid answer too! 773