All children are different. I was very careful with knives, even from an early age, because my parents told me to be (I was very obedient). My oldest nephew is 10, and has a knife, but I think they waited until he was 8 or 9 before giving him one, because he's less careful with things. I know some adults who are less trustworthy with knives than some children.
Ultimately, it's up to the parent to gauge when to give a child a knife, based on that child's personality and maturity level. When I eventually give my grandson a knife (he's 3 right now), it will only be with the permission of his parents, since they are the ones who are around him all day. I'll teach him knife safety and such, but they are going to be better judges of when he's ready to accept that responsibility.
Accidents will happen, and I wouldn't be too concerned if a child cut his hand with his knife while learning to use it whittle or shave wood. Pain is a good teacher, as I'm sure many of us can attest to. I would be more concerned about him waving it at other people or animals, throwing it, running with it, etc. Accidents are one thing - immaturely brandishing is another thing altogether.