Same thing happened to my wife's phone three times actually. The first time I just thought she did not plug it in right the night before. Second time I though there was a problem so off to Verizon I went. They were alot of help "NOT" and said they needed it for a few hours to test the battery and charger. All fine and good but at the time I couldn't leave it so I went home and had customer service walk me through a hard re-set. The third time it happened while the phone was on the charger and I was able to get a replacement.
All three times it was acting the same way, showing 5% power until you plugged and unplugged it then it went to 50-60% and charged after that. There was no rhyme or reason to it and it didn't have extra apps running in the background. The killer was when it discharged while on the charger.
"He who wrestles with us strengthens our nerves and sharpens our skill. Our antagonist is our helper". Edmund Burke