Stuart is correct. Ohm's law is not being broken.
Trying to cram a large amount of current through a wire that is undersized will create heat on the wire.
Heat is a sign of resistence, which makes sense, seeing as not all of the current is getting through to power the lights. The current that is not getting through to the lamps is converted to heat energy. This is basic high-school physics. Have a look at the law of conservation of energy some time.
As a point of interest, this is the way that
Rheostats work (variable resistors, like a dimmer switch on your dash lights or the fan switch on your heater). They use special wire called resistence wire that drops the current down to the desired level, but in order to satisfy the law of conservation of energy, there's heat generated. The higher the resistence, the more heat is outputted, and the less current is passed. If you've ever touched the back of a rheostat switch like that when it's been on and burned yourself, you've experienced this physical law first-hand.
