I have two 7.2V 2.4Ah Li-Po batteries that both have this problem. They have passed down a few channels in the Navy so I know little about them, but they look like unused custom batteries (I was also told they were good batteries). No datasheets. Upon visual inspection, looks like 2-cell, in series. I was told that it has built-in balancing electronics (but he didn't sound 100% sure). I don't know charging/discharging rates.
I charge them individually using a Dynamite Vision Peak Ultra Battery Charger
(on the Li-Po setting). I've tried both 0.5A and 0.65A charging rates. I don't remember exact times, but both charged for about ~1.5 hours, then after waiting a day, I charged them again for another ~30 min.
Both the charger and my multimeter confirm the battery finishes charging at around 8.2V. But here is where it gets strange . . .
Attaching a simple LED/resistor combination, the LED lights up for a second, then quickly fades away. My multimeter reads ~1.4V with that plugged in. Unplugging the LED, voltage jumps right back to 8.2V. This is 100% repeatable.
Attempting to run a servo results in a ~0.4V reading. Unplugging it, back to 8.2V.
Either the charger isn't working, both the batteries are faulty, or I'm doing something really dumb . . . The charger works fine for NiMH batteries (on the NiMH setting, of course).