Obviously, heavy spoilers for Harry Potter and the Deathly Hallows, do not read if you don't want to be spoiled.
So they find the locket horcrux, and before they figure out how to destroy it, they decide to take turns wearing it. Why on Earth would they do that?
We know (and they quickly figure out) what influence it can have on one's mood, and it eventually leads Ron to give up and abandon Harry and Hermione. Hermione, on the other hand, has a wonderful bag that can take whatever you put in it. Why not just put the Horcrux in the bag? Do we have any information in canon that indicates the Horcrux would have still had such an impact on Hermione's mind?
It seems a pretty suicidal behavior to me to wear a Horcrux which basically makes you weaker, especially if you're on a path to destroy such Horcrux.
While there are good answers, I feel like I cannot accept one of them yet due to Anthony Grist's first comment to this answer which seem very relevant:
The main thing that bothered me about the whole thing is that Harry has an entirely secure bag hanging around his neck at that point. There's no way he could have forgotten about it since it would have been right there when he hung the locket around his neck too.