Hi, I have a late 2011 Macbook pro and I keep it plugged in most of the time when I'm at home. Sometimes the power adapter's light doesn't come up but the mac indicates that it is still plugged in. If I unplug it and plug it again then the light comes up and becomes green.
My spare charger has been doing this for about a year, I've just put it down to the LED/other non-vital component failing in the MagSafe charger. Hasn't stopped the computer charging and seems OK otherwise. Out of interest, what MagSafe is it? The faulty one I have is an old white 'T' connector, my other 3 metal 'L' connectors have been fine.
Thats what my charger looks like. Battery level has always been at 95% and shows as plugged in. But the megasafe light is off now.
If it doesn't do it with any other charger then it's not your computer. As long as it recognizes the charger and is working properly minus the light, then you're fine. Happened with one of my old chargers.
Use a wooden toothpick to clean around the contacts on the connector. Might just be a bit of schmoodge.