Some can, many can't. Unless the split is mutual, there's really no reason that you should have to remain friends.
Think about it: usually it's the dumper who wants to remain friends, because I guess he thinks that will make him seem mature. But, really, why should the person getting dumped feel the need to stay friends?
The dumper has probably been thinking about it and getting used to the idea of not being together for a while, and oftentimes the break-up comes as a surprise for the one being dumped.
The person being dumped might even be shocked by it, and now he's supposed to say, "Well, I'll be the bigger person and stay friends with my ex." But there's no reason for that.
He just got dumped by someone he probably still cares for like a boyfriend, and now he has to play the friend--listening to him talk about how he's moved on to the next guy. The guy who dumped him was ready to move on, but he wasn't. There's no honor or being the bigger person in putting yourself through that.
Maybe after some time passes it can work, but I personally think that if you're about to break up with someone--especially someone you know still cares for (if not loves) you--you should be prepared to accept the fact that he's entitled to resent and even hate you. That's just a price you have to be willing to pay when you dump someone. You can't have the best of both worlds.