A simple search on Apple's forums shows a common trend: a lot of iPhone and iPad owners are having problems charging their device. Two of my colleagues have also run into this problem. There's a simple solution to this. Before blaming the cable or phone, check your device's Lightning port.
Lint and other debris from our pockets or purses gets stuck in this port and can build up over time. To be safe, back up your iPhone or iPad to your computer or the cloud. Power it off, and with a normal toothpick, gently remove the lint. You will be amazed how much can get stuck in the port.
Connect the charger and see if it works. Still having problems? Try cleaning the port out again, but this time use a can of compressed air. Alternative options for toothpicks include using a SIM-card tool, a bobby pin, or a small needle.
You can also use this method if your iPhone or iPad is stuck in "headphone mode" even though no headphones are plugged in. The same method can also be used for cleaning out the charging port and headphone jacks on Android, Windows Phone, and BlackBerry devices.