I'm seeing this all around me, and within me, too.  On the treadmill, on the weight machines, in the food diary... going so far only to give up because you "can't do it."  Two things.

1) Melissa always taught me to never say, "I can't" in the gym.  It's not a good mantra to have in life, and it's a terrible one to have in the gym when you're battling a life-or-death situation.  "I can't" puts invisible limitations in your mind that your body will listen to.  Fitness and health is twice as much of a mental battle as it is a physical one, and I 100% agree with that.  If you're on the treadmill (like I was today), and your feet and legs hurt (like mine did), telling yourself that you can't finish the mile will most likely translate into you NOT FINISHING YOUR MILE.  But, if you tell yourself to keep going and that your body is capable of finishing, guess what?  You'll probably finish.  And be all the better for it.

2) Don't take this meme to the extreme.  If you're really in pain and really hurting (like the kind of hurt associated with an injury and not just a tired body), then stop.  There's no need to prove to yourself or anyone else you can do the workout despite being injured.  That's dumb.  But, if you take this meme at its most basic level, it's telling you: "Don't quit.  You're already sore and sweaty and tired.  You might as well get something out of everything you've put in."  That's my take on it, at least.  And think about it, if you push out those last 5 reps and go until you burnout, you're doing the most good in those last 5 reps that you did the first 20.  That shaky, sore, and lifeless feeling you're fighting is building strength and endurance that will help you do 5 more the next time.  You're putting the work in-- why not take it for all its worth and reap the benefits?

Unless you're injured, keep going.