I just had the same problem on an iPhone 5 that's 7 months old. Same problem as most, while the phone shows that it is on charge, it never charges. After unplugging and plugging back in several times, I would occasionally receive the Non-Apple message. Fortunatly i have several cables, and tried an alternate, which began charging almost immediatly. I also carefully inspected the charge connecter and found NO lint whatsoever.
I then tried the original faulty cable, and by applying light pressure, found that it would alternate between charging and not charging.
I believe there are a couple of distinct issues here. The connection port getting filled and thereby preventing adaquate contact. And secondly, faulty cables.
Whatever you do, don't mess around with switching batteries as mentioned by a "Phone repair" person earlier in this post. Do yourself a favour, use a flashlight to see if there is any visible lint in the charge port, if so, carefully clean it out with a small pin. Avoid touching the connectors, simply pry out the lint from the sidewall. If there is no visible lint, replace the cable. If it's still under warrenty, stop by the Apple store and "Demand" a new cable.
I hate to say it, but i think the design of the lightnin adapter has introduced higher tolorances, which unfortunatly, the cables don't seem to live up to in the long run. For all we know though, Apple may already have addressed this issue in the recently produced cables.
Good Luck!