Welcome to Android Central! Are you plugged into a wall outlet or a computer USB port? Computer USB ports don't supply much current, so if you're actively using the phone, the battery will often still go down while plugged in.
What are you doing while charging -- playing a CPU-intensive game? If you're doing something that uses a lot of battery, it can still outstrip charge rate.
Also, the Note 3 is very old, so the battery is probably pretty worn out. You can look into a replacement battery, but chances are any replacements out there are also very old, and therefore may also be worn out as well.
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