Perhaps the most important trait to have when hunting for non-profit jobs is thick skin. Like it or not, you are going to be rejected. In fact, this will happen more often than not; it's just the nature of the game, unfortunately. It's never fun to get those job rejection letters, and it's even worse knowing that they were probably copy-pasted from a meticulously written sample. If you are going to eventually find yourself in the nonprofit sector, you are going to have to be deal with the sting of rejection, get back on your feet, and apply for another position.
I am quite aware this is easier said than done. My first few times getting the dreaded rejection e-mail (or just not hearing back at all) were some of the hardest job experiences I had to deal with. It's almost akin to The Stages of Grief: first you get angry, then you get depressed, and you wonder if you're ever going to find a job. But sticking with the grieving process analogy, it's important that you get to that final step: acceptance. Yes, it's awful that I wasn't chosen for that job I really wanted, and yes, it's important to acknowledge how much it hurts. But it's also important to remember that being rejected for a job is not an indictment of your personality. More often than not, the reason you are not hired is because the employer found someone who is a better fit than you are.
But the most effective job rejection response is to simply apply for another job. Let's face it, you're never going to get a job if you just wallow about the one that got away. The nonprofit industry is moving fast and if you don't get back on your feet after a setback, you are going to be left behind. So get back on your feet, weary job hunter! Opportunity isn't going to come to you, after all.