Huh, we just had this exact problem, except in our case, the machine was running fine, plugged in, and then just sorta started draining battery. Replugged cable, it came back up and started charging... Briefly. Then died again.
Since I have a nearby laptop that can check whether things are providing power, I've verified that the adapter is working at least well enough to provide 60W (I wouldn't expect the Dell to see the 85W mode). I also have a Dell dock that works and puts out 60W, and an OWC dock that appears to put out 45W, and all three work with the Dell, and none of them appear to make the Mac charge.
Machine originally arrived around... I think November 15? Been in pretty regular use since the 17th or so, no problems to speak of before today.