The filter in the dryer catches quite a bit of the lint but not all. Over a period of time some of the lint gets into the vent pipe. If there are any restrictions it will start to clog it up until it is very badly clogged. A clogged vent pipe should have caused the thermal fuse to blow and not short the coil.

I totally agree with the fire chief! Not enough people know that although the lint screen being cleaned after each cycle is important, not only so the dryer can work properly to have the right air flow throughout the entire system, but because of the build up of lint becomes a fire hazard. Going beyond the lint screen should be an annual maintenance job on your household checklist. Anything getting past the lint screen will catch inside the vent hose in its accordion-like folds. Once it gets clogged, you are then pumping that heated air from the dryer into it, asking for trouble. Sadly, many unsuspecting people have had major damages to property due to lacking this very knowledge.
Another thing you can do as a monthly maintenance item is look at the lint filter itself. Is the screen clogged up a bit? This is because we use dryer sheets and in-wash softeners to make our clothes soft and fluffy, just like the teddy bear promises. The make-up of dryer sheets causes the softening agent to come off in the cycle and go into the fibers of the laundry. Liquid softeners leaves a residue on the load, so when you transfer it to the dryer, it does its thing there too. Some of this residue also ends up on the lint screen. After time, it's not so much a filter anymore, but a deflector. A lot more of the lint will get through to the vent hose if your screen is clogged, because the air is blowing against a more 'solid' surface than a screen.
A simple fix for the lint screen, without having to give up your soft clothes, is to use a semi stiff brush and hot water with a good detergent (like dish detergent - Dawn or similar) to clean the screen. Don't scrub too hard and make holes in the screen, but enough to clear the gunk off. Let it dry before you put it back in. I find wrapping a towel around it and patting it dry does wonders.
This is a win/win for you! Your dryer works better/lasts longer, and you don't catch the house on fire with your dryer lint... Saves money all around, and keeps the wife happy!
Hope this helps anyone that didn't already know.