Is it possible to move on from a break up and still remain friends with the ex?

You care about him, but he's taking advantage of you. Your care for him may have been due to him presenting well enough to care for, but deep down, he is being selfish in not letting you go. It's unfair that you feel such commitment to him and that he isn't allowing that love from you to be enough to either commit or let you go; it's cruel, in fact. 

Imagine that I tell my job I will come in to work, but then don't show up and then show up the next day and work really hard when I need the money. Imagine that work would have to hold an entire spot for me even though I don't commit to them. That would devastate the business. Much worse is happening inside of you as you accept this behavior from him as acceptable in your life. This is what is happening. Is this how much you value yourself, to be treated as convenient? It's not some noble thing on his part; it is immature selfishness, and you need to respect yourself enough to not deal with it. 

Unless (this part is important) you see characteristics inside of him that are worth investing in. If you see a young man, not yet mature, or a good, caring person for others, who thinks of others' needs before his own, and is confident (not cocky, that is ego), then that tells me he might be worth investing in. However, that I believe to be rare given he is willing to string you along, as you present. 

What is worth investing in is the man who says, "I am not good for you; we are breaking up." A weak man can't do that. A weak man tries to hold on to you while still open to exploring other people. I fear that he fails at a new relationship and then comes back to you. I fear that he gets the attention of another girl and then strays and then comes back when she figures him out. That is the sign of a weak man; he goes wherever the wind blows him, he is unsteady in himself and needs external validation and reassurance of other women. 

What you can do is attempt to see this situation for what it really is, a young boy not yet man enough to be with you. He isn't a BAD person or someone who is evil, but he is immature, thinking he can play a relationship like that and get away with it. 

I don't know this man's history, but if he has a history of childhood inconsistencies or trauma with his primary caregivers, then he may be struggling with a deep sense of inadequacy or being a victim, which will result in sad stories for other people to receive the reward of validation. This would affirm what he cannot do for himself. He cannot support what he doesn't accept, which is all of him, the inadequate parts as well. If he cannot accept these parts, he will hurt you because he doesn't believe you will stay, and so he will push you away before he gets too hurt. He's keeping you at arm's distance now in fact. 

Your job is to ask yourself what is good for you and what you are willing to put up with. If you are willing to put up with his behaviors, then commit to it. Commit to the feelings of being wronged or treated inappropriately. If you really commit, then all of these thoughts and feelings come along with the decision. Either commit to this chaos, to else you better remove the negative and troubling from your life the best you can; that starts with him.