Not charging when battery gets around 10-15% Remaining

I bought this Macbook Pro second hand. The guy told me the computers works great but there was issue that would happen when the computer got down to 10-15% in battery life, it didn't want to charge. I took it by the apple store for a second opinion. I noticed on the way in that the cord he gave me with it was frayed on wire going into the wall. I was prepared to purchase a new charger, because that might be a safety concern. I let the mac specialist look at the cord, and he decided to try another one. He ran several diagnostic tests on the computer. The battery checked out perfectly, in fact it hadn't even ran through 20% of its life cycle count. He was still experiencing the same issue with his new charger. He then explained his best guess might be the charging port, and said they could fix it for a fee, but it would take 5 business days to get it back. I offered to buy the part from them, however they apparently do not sell parts to customers, but he did give me the part number. He then gave me a brand new retail box charger for free, which was very nice of them. I have not changed the part out, because I wanted to see if anyone might have a solution beforehand.

The battery charges great but when it gets to that percentage, the light on the charger goes out. I have to unplug the charger from the wall and wait a few seconds, then it will go back to charging, but for just a few minutes. Once it goes over 15-20% it continues to charge until full. The Macbook runs smooth at all other times, and never gets hot, charger is not getting hot either. Any ideas? Is it a bad mag safe board? or something else?