I think that is very possible to stay friends with ones ex and I believe many people do it willingly, I however have not remained friends with any of my exes but I could and I know and hear of people who do especially when they have children. Some remain good friends since they know each other really well and they find it easier to confide with them.
I always found it weird when people stayed friends with their ex. Whether it was a bad falling out or not, why would you want to hang out with them still? If it is over and done with just move on with your life and be happy. There is no reason to hang around an ex.