I realise this is extraordinarlly unlikely but is it physically possible? If a bullet entered an empty barrel at just the right moment as it moved around to fire again could it feasibly jam the mechanism?
(I’m not sure what the difference is between a gatling gun, mini-gun and chain-gun but the question is applicable to all of them)
Do not need answer fast.
Well, sure it could. A Gatling has multiple barrels, the original had 10. Every barrel is in a different state of readiness at one moment, from empty to loaded and uncocked, to partially cocked, to cocked a little more (it takes a while to fully cock as it rotates up the cocking ramp) and finally, bang. I have the blueprints to build one but haven’t found a year to spend yet.
Exactly what would happen would depend on what state that barrel was in. They are cantankerous old beasts and it doesn’t take much jamming to lock one up. An original Gatling was 45-70 caliber so even a .40 round could miraculously go down the barrel.
Because of the design, they are a pain to unload. The one I have plans for has an decocking lever you can throw and the firing pin no longer gets pulled back, but there are already a few rounds past that point. You turn the lever, remove the magazine and crank until it stops firing firing and ejecting brass.
Dennis
I agree. It really is incredibly unlikely, but, if it happened, a bullet going in the wrong direction could easily cause firing chamber destruction or even complete barrel failure. It doesn’t take much barrel blockage to cause a failure. Dirt, small rocks and even snow can cause failure if you fire with such a blockage. It can and does happen in real life and not letting that happen is a key gun safety warning. Even simple obstructions can cause an instant overpressure that just just destroys the whole thing possibly injuring or even killing the shooter in the process. A bullet coming in the opposite direction is a lot worse than some mud stuck at the end of the barrel.
Here is a real-life example:
The Gatling gun was a hand-cranked gun with multiple barrels, first made during the Civil War. Modern rotary cannons are often called Gatling guns or Gatling-style guns if they have multiple rotating barrels, but that’s kind of a slangy use of the name Gatling. The modern guns are more properly called rotary guns or rotary cannons.
Similarly, the minigun was originally a specific weapon made by General Electric (the M134), but like the Gatling gun, the name is now applied generically to any rotary gun that fires regular rifle rounds or rounds of a similar size.
So for example, while the A10 Warthogs huge GAU8 gun might be called a Gatling gun, it’s a little bit too large (understatement) to call it a minigun.
A chain gun is a machine gun that uses an external source of power like an electric motor to cycle the rounds. This is in contrast to other types of machine guns that use either the recoil or gases from the last fired round to cycle the next round into firing position. It’s called a chain gun because many of them over the years have used a crank or an electric motor with a chain attached to it which operated the gun.
So a multiple-barreled rotary gun that was driven by a chain and motor and happened to fire rifle-sized rounds could be called a Gatling gun, a minigun, or a chain gun, and all three would be correct. Well, correct-ish, anyway. It’s kinda like calling any sort of tissue a Kleenex, even if it’s not made by Kleenex. You’ll find people who insist that it’s a Gatling-style gun and not a Gatling gun since only the Civil War version was a proper Gatling gun.
Thanks for the answers everyone, that was very comprehensive and informative in particular engineer_comp_geek 
The same issue is equally true of any firearm / cannon ranging from a .22 derringer to the WWII German Schwerer Gustav - Wikipedia.
The multibarrel-ness or not is totally immaterial. If some junk gets partway down the barrel of any projectile weapon then the next time that barrel fires a bad outcome is almost assured.
Recent example from non-machine gun: JeffCo deputy's "one in a billion" shot went up barrel of suspect's gun during Aurora shootout
Wow! What are the chances of that?