The battery icon doesn't flash and the iPod doesn't charge. In fact the opposite happens and the battery drains as the iPod won't actually power down. I recently noticed this as the iPod battery was almost flat after being left connected to the computer overnight (which was off, but not unplugged).
Could it be that there is actually a trickle of power being delivered which keeps the iPod from shutting down, but which isn't enough to cause it to recharge?