My Macbook Pro and Magsafe power adapter are fed up with each other.
All the lights light correctly and the Mac battery receives a charge when I plug in someone else's adapter, so the problem is not my Mac.
My Magsafe adapter works correctly when plugged into someone else's computer (all the lights) so the problem is also not my adapter.
I have a friend who has the same problem, and when it happens we can trade adapters for a while and it seems to fix the issue. I'd love to hear a technical explanation of what might be going on from someone familiar with how these batteries and chargers interact.