I am not a pilot - merely a keen aeromodeller who uses lithium polymer (li-po) batteries to power radio-controlled model aircraft. We have taken an old, smallish (3 cell, 2200mah) li-po in a clapped out, discharged state (3 volts a cell approx, compared with 3.7 volts nominal and 4.2 volts fully charged), placed it on a brick in the middle of a field, shorted it out and run off.
The result is a battery which puffs up and then generates a phenomenal amount of smoke with significant charring to the wet grass around the brick. So much for clapped out. Catastrophic in a plane I should think.
Most of the incidents with these batteries in the aeromodelling world appear to be caused by faulty battery chargers/electronic component failure in aircraft or human error in handling (dropping batteries, crashing planes etc). We never leave batteries unattended when charging and whenever possible charge in the open air.
Some time ago the AAIB reported on batteries which had caught fire in a full-size glider which had an electric motor, fortunately on landing if I remember correctly.
So, items to be handled and stored with the greatest of care. My laptop and mobile phone haven't caught fire while I have been writing this, fortunately, but their batteries may be lithium ion about which I know little.