1. You are in a job that was supposed to be a dream job but in fact you hate it, you make little, and your boss does nothing but yell at you. Do you want to stay and continue being abused, not growing in your career, and possibly suffering from its affects on your personal life?
2. The guy (or girl - I'll be fair) you were dating never initiates contact anymore, barely responds when you do, and claims to have no time for you, yet you hear stories from others about him (her) being out with his (her) friends. This actually doesn't even have to apply to just romantic relationships, but friendships as well. Do you want to keep waiting for the other person to return your text, ask to hang out, show more interest, or do you want to walk away and, though it may hurt, realize that you deserve better than that?
3. You are doing what you thought was a passion (job/interest/hobby/etc), but it turns out that you're not headed in the direction you'd thought, and really aren't enjoying it much any more. Or worse, you know you don't enjoy it but can't seem to let it go. You feel trapped because you feel like you "should" keep going but when you trust your intuition and gut, find yourself being pulled in another direction. Does it make sense to keep entrenching yourself in a situation that's so unrewarding?
Each of these is an instance in which, depending perhaps on some other personal factors, you may actually be better off "giving up". The reason for this is that you actually aren't giving up. Rather, you're realizing when a situation is bad for you or is no longer in line with your goals/dreams. In fact, these could be holding you back, not only having a negative effect on the day to day, but perhaps on your self-esteem and your belief in yourself. If you take a step back and analyze these situations at their core, it's possible that you're staying with them simply because you don't want to be a person who gives up. Sometimes, though, it's ok to give up. In fact, it can be healthy.
Obviously, you need to understand the consequences of your actions. If the giving up involves a person, you may have to give them up for good, or at least a decent amount of time. Not always, but perhaps. If it's a job, you may have to give up that experience as a reference, you may not be able to go back if things don't work out down the road. But sometimes, even these consequences are for the better. In "giving up" here, we're respecting ourselves enough to admit that we deserve better, and that we deserve happiness. Because in fact, we're not giving up at all - we're moving forward.