I have a late-2008 15" Macbook Pro, model A1286. I had many of the symptoms people are describing in this thread...battery wouldn't charge, the power cable had no issues, the charge light on the power adapter would intermittently show Green but not charge, or not light up at all.
I replaced the battery (it was time anyway, wasn't holding much of a charge) and that didn't resolve the issue. I borrowed a power adapter from someone, that wasn't the problem. It turns out to be the MagSafe DC-in board, a small circuit board inside that contains the Magsafe plug. I don't know which other MBP models have a separate DC-in board, but I do know some newer models have it contained on the Logic Board, which is a much more expensive fix ($600 for the LB, vs. $30 for the DC-in board).
The MagSafe plug fails because the entirety of the force from plugging and unplugging your power chord is being absorbed by two small weld points on the circuit board. You might thing that all that force from the magnet when you're plugging & unplugging is being absorbed by the aluminum body, but it's not. The end of the power cord forcefully connects to the grey area surrounding the actual power connection on the side of the computer, and that grey area is only held on by the two small welds inside. It's not hard to imagine how this would eventually wear out, begin to put strain on the actual power connections, and before long you have a failing power port.
I bought the replacement DC-in board from ifixit.com for $30, but be very careful if you're going to do the repair yourself. You have to disassemble the entire computer to replace this one component, and if you don't have any experience in this area I suggest you find help. There are many small, fragile connections in the computer that would be very easy to permanently break if you're not careful. I believe ifixit.com has a thorough walkthrough for this issue.
Apologies if anyone has mentioned the MagSafe DC-in board already, I scanned through the thread and just saw a lot of people who couldn't figure it out. I hope this helps someone.