Is it possible for a mechanical failure to occur that would result in the hammer being dropped? Like a broken part or something. Simple yes or no will do, but explanations appreciated.

Yes, an example would be the hammer hooks and sear interface failure due to wear or improper stoning/polishing. Charles.  

Yes, but not likely.

I have a BHP that I've seen the hammer follow the slide three times. It get caught by the half cock notch but that's no excuse. I suspect a worn sear but won't know until I tear it down.

If you have a worn out firing pin spring and if you have very soft primers and if you dropped the gun directly on its muzzle on a very hard surface from the top or a building, it might have enough inertia to dent the primer enough to fire the gun.  

I really can't imagine another type of failure that could happen.  Even a hit to the back of the hammer would have to break two sear hooks plus the "half cock" notches before that would allow the hammer to hit the FP.  

Firing pin sfeties are pretty useless on a 1911.  I don't know why manufacturers insist upon putting them on new guns.

Sure,

Hammer following the slide down and bang. Doesn't have to be a broken part, just not "adjusted" properly. Seen it done and read about it more than several times.

Not common with a properly/fitted gun however.

Quoted: Sure, Hammer following the slide down and bang. Doesn't have to be a broken part, just not "adjusted" properly. Seen it done and read about it more than several times. Not common with a properly/fitted gun however.

Hmmm... well the Kimber style FPS wouldn't stop that unless you dropped the gun to release the grip safety and therefore, the FPS.  On a Colt or Sig, you would likely still get a double, but could prevent a mag dump by releasing the trigger.  I hadn't considered that.