Ever since Wolverine's origins have been explored in the comics, he has always had bone claws first that were later coated in Adamantium through a painful process during his time in the Weapon X program.
It has also been shown in the comics that under extreme circumstances and through excruciatingly painful events, the Adamantium can be torn from his bones, leaving him with bone claws again (which, incidentally, can be broken off and in fact, have been broken off. Also, extracting the Adamantium caused several other 'reactions' within his body such as his healing factor showing its full potential and an 'increase' of his mutation to where he turned more bestial).
Shouldn't there be a thread about the Origins movie around somewhere though where this question would fit better than in the thread about an upcoming X-Men related feature?