MacBook only charges with non-OEM chargers?

Having a very bizarre problem with my machine. It only charges with non-OEM charging cables. I have an OEM 60w and OEM 85w charger and when I plug either into this machine, I get nothing. No green light or orange, system profiler doesn’t detect being plugged in, and battery drains as normal. I have other machines which work fine with these cables so I’ve eliminated the possibility of the cables being at fault. However, I’ve tried 3 different 3rd party cables, both 60 and 85w, and they all work fine! I don’t understand it, and any advice is appreciated. I’ve already ordered a new DC magsafe board and I’m hoping that replacing it will fix the problem, but any other suggestions or even an explanation would be helpful. I’d much rather use an Apple charger than a sketchy 3rd party one. Thanks!