If it's very old, this seems to happen. Old batteries can also damage the charging circuits and chargers so they symptoms match what I've seen in old battery failures.
Bob
My MacBook (13" Late 2006 2.4 Ghz Intel Core 2 Du, ) running OSX 10.7.5, will not charge the battery when the computer is running. Shutting the computer down, waiting a few seconds the charger will then charge the battery.
In addition the computer will not "run" on the power adapter alone (i.e. with the battery removed).
I have tried SMC and PRAM resets.

Chowhound
Comic Vine
GameFAQs
GameSpot
Giant Bomb
TechRepublic