Marvel.com
Wolverine’s skeleton includes six retractable one-foot long bone claws, three in each arm, that are housed beneath the skin and muscle of his forearms. Wolverine can, at will, release these slightly curved claws through his skin beneath the knuckles on each hand. The skin between the knuckles tears and bleeds, but the blood loss is quickly halted by his healing factor. Wolverine can unsheathe any number of his claws at once, although he must keep his wrists straight at the moment his claws shoot from his forearms into his hands. When unsheathed, the claws are fully within his hands, and thus Wolverine can still bend his wrists. The claws are naturally sharp and tougher than that of normal human bone structure, allowing Wolverine to cut through most types of flesh and natural materials.

Now show me the proof this is all wrong and I will believe it. However, even if in the early comics the claws were not made out of bone, they were in the ones following, there are many alternatives in the comics because there was always different comic routes. That being said I still want to see proof that his claws were added by X.

Wolverine vol 2, issue 75.
After Magneto ripped out Wolverines adamantium skeleton it was revealed that Wolverine had claws made of bone, meaning he was born with them. He even says at the end of the comic that he must have been born with them, but he can't remember having them.

Wolverine and Gambit, issue 85 and 86, The origin of Wolverine,
he uses them to kill Logan back when he was young, way before X.

The kid is actually WOLVERINE, and this claws are not adamantium yet, those are his bones. HE IS BORN MUTANT. YOU CAN PAUSE the part of the trailer where his claws are shown, the claws are not smooth. he only got his adamantium claws when he was experimented on and there he got his metal claws.