Repaired my sister's grandson's Tab 3 which had the same problem, won't show it's charging, won't power on. You have to remove the back cover of the tablet and disconnect the battery cable from the circuit board, then reconnect it back, and put it on a charger. The display will flicker/pulse when connected to a charger, and after a while will show it's charging because the battery was ran till dead. My advice if you're using a mobile phone charger, generic charger, or computer usb port is soon as you can power it on turn off the wifi and then turn off the screen (hold the power button till the screen goes dark) but if you have a 2.1 amp tablet charger you won't have to do either and can leave it as is since it can supply more power than the tablet is using (can't say that about the prior three). I posted a youtube video about this on another post.
I just had to fix my sister's Tab 3 that I bought her for Christmas by disconnecting and reconnecting the battery. It's ridiculous that they won't address this issue that's been around for at least two generations. This is why every battery should be easily accessible.
In honesty this is something basic that tablets lack compared to a laptop. If you simply had an app/option to lock the device or power it off at 10% power, this could perhaps be avoided permanently. I still have suspicion that some issues are more common on the model with PXA986 processor, and could be resolved by a firmware update that changes the battery charge calculator.
And the likelihood that Samsung will provide one of these prevention mechanisms via software update is.....????
My daughter's Kids Tab 3 is having the same issue. My two sons each have an 8" Tab 3.0 and have not had this problem. When I brought the Kids Tab back to Best Buy two weeks ago they connected a charger to it and a few minutes later it woke up out of its coma. Today is seems to have re-entered its coma. I can't charge it with ANY of the ORIGINAL Samsung chargers I have (the one that came with the Kids Tab, the ones for my sons' 8" tablets, the one for my S4 phone, and the one for my wife's S3 phone. And it won't trickle charge from my PC. All 5 of my SAMSUNG chargers work just fine with all of our Samsung devices except for the Kids Tab. The Geek Squad at Best Buy must have some "magical" charger because it wasn't even a Samsung charger. Looks like it's back to the store for me again. If they replace it (which the likely will because of my service plan) how can I prevent this from happening again? Maybe I can buy that charger that they used???
I have the same problems with my 3.0 tablet. The only way i have seemed to get it to work is to alternate between a usb connected to a computer and the wall mount. Let it charge for about 30 mins and try turning it on (hold power button for 10+ secs). If it doesn't work in one power source switch to the next power source. Eventually it turns on. Make sure the device is plugged into a power source when trying to turn it on. Its frustrating I know, but eventually I manage to get it booted up.
Please let me know if this works for anyone else...