I am currently having the same problem with my iPhone 6, but there is no worry. This is a common thing to happen with all iPhones. What I recommend doing is charging the phone until it turns on. Then, even if it shows it is not charging, power your phone OFF, and keep your charger plugged into the phone. I did this at night and when I woke up my iPhone was at 100%. If this didn’t work for you, you can go to apple customer service (online) and talk with a apple tech. They will help you with any problem you have.