I don't see myself as a quitter, so occasional I struggle with leaving someone or something, even if the reason seems good and valid.
How does one determine when the appropriate amount of time and effort has been put into an unresolved endeavor or fruitless situation before being able to responsibly say 'it is time to put this aside?'
Unfortunately, I think that each situation requires scrutiny and that there is no complete answer. There are situations that one could and even should stop almost immediately; and there are trials that one can never stop, no matter how painful or frustrating they become.
Pain is not a reason by itself to stop a course of action, but it is something that should make you ask questions. If your ankle begins hurting when running, you slow down, assess what is causing the pain. Maybe you stop for the day, but you don't give up running for life. However, if your knees ache every time you run and it never gets better no matter how hard you try or how many doctors you go to; well, maybe it is time to take up cycling.
I believe that a single, or even multiple failures does not mean throwing in the towel. If that were the case, I could never be a Christian. In a sense, the Christian life is about accepting failure: My failure at purity, at holiness; my failure to redeem myself; Humanities failure and fall in to sin, our races complete depravity and complete inability to remove ourselves from it. However, accepting those failures does not mean you don't strive for holiness. Christ's sacrifice gives us the power to become more like Him, the Spirit works in us to achieve that which we couldn't on our own. We will still fail because our dying sinful self fights as it dies, but that doesn't mean we should ever give up, even though we will not have full victory in the life.
The confusion comes with anything that is not as important as our increasing Christ-likeness, which is everything else! How do I know when a situation is beyond me, when it has become unhealthy or impossible to succeed?
I do believe that it is too easy to give up. Struggle and failure is pain, and pain is unpleasant. Of course we should want to avoid it, end it. However, not all pain is bad; the trick is learning whether the pain is a pain that will lead to growth or a pain that acts a warning. Is this a pain of God stretching and increasing Himself in me, or is this pain God's megaphone? Do I keep running or take up cycling?