I had a similar problem which only seems to have arisen since the kitkat update. Tab charged very slowly or not at all when on it's own charger. 25% to 32% over night. Disconnect and reconnect, and it would charge a bit more. Tab charged OK if a bit slow from USB3 port on computer.
Tried a factory reset, using kies for back-up, no joy.
Thought it might be the charger since the tab could be charged from a computer USB port. Bought a new Samsung charger and cable same as the one that came with the tab, no difference, same problem.
Finally sorted by wiping the cache partition - here's how:
1.Turn off the tab completely (not just sleep mode) by holding down the power button until until you get the Device options menu, select Power off, then press OK on the Power off menu that follows.
2. When the tab is completely off press the vol-up button and the power button together and hold them down. When "Samsung Galaxy Tab 3" appears on the screen, release the power button but continue to hold the vol-up button.
3. An Android character lying on his back will appear with a menu over. Release the vol-up button.
4. Use the vol up/down button to scroll down the menu choices and select "wipe cache partition"
5. Press the power button. When it has erased the cache select "reboot system now". You should now be good to go.
This solution worked for me and is worth a try. It's quick, doesn't cost anything, doesn't need any tools, avoids the hassle of a factory reset, doesn't involve sending the tab back to Samsung and won't void your warranty.
The cache is sort of scribble space that the operating system uses to speed up frequent operations. It can get a bit cluttered with old stuff particularly when the OS has changed as in the case of the kitkat update. The system will rebuild a new cache and you won't notice any difference other than (hopefully) the disappearance of the problem.
