Usually no, especially not immediately after a breakup. I’m Facebook friends with an ex, but years after we broke up (Facebook didn’t exist when we dated) and I’m still not “friends” with him really. I’m not in touch with him like a friend, but we are friendly. I like to see how he is doing, his kids, I “talk” to his wife more on Facebook status than him.
I think people can be friends once both have moved on completely from the romantic relationship.