If this happened to you, I may have a possible solution:
1) you let the battery die all the way
2) you are using a non-factory charger
I just went to Samsung website and the first thing they ask you to do is not try to charge with a non-factory charger.
Well, I went through my mountain of chargers and found the white travel charger that came with the tab 3, the moment I plugged it in, wammo, it started to show charging icon. Once I let it charge for a couple of minutes, which brought it to 5 percent charge, I switched to my multi port non-factory charger and it kept charging without any issues.
Lesson learned:
1) don't let your tab 3 drain all the way
2) if you do, you will need the original travel charger to bring it out of its charge coma.
Hope it helps...