Step 1. Don’t use Hoppes Bore Snakes or similar items.
Step 2. Don’t pull harder, you’ll just make it worse.
Step 3. Get USGI steel cleaning rods and attach enough sections to reach the stoppage.
Step 4. Heat end of cleaning rod to one hundred billion degrees.
Step 5. From the chamber forward, push super heated rod into the stoppage and slowly twist until it passes through. You’ll feel the pop.
Step 6. Pull Bore Snake out, throw in trash with the tofu your wife bought.
What size bore snake did you use? What caliber was the rifle?
More important than the 4 firearms safety rules! 
If you had used zip ties this all could have been avoided
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Step 1 ruins all the fun.
This contains much wisdom across manifold subsections of life.
I see what you did there lol
Well, considering it’s an 11” HK416A5 upper, one would surmise that as a SME I would and did use a 5.56 Bore Snake lol
I believe it suffered some material failure (though it was pretty new) which caused a tension knot where the bristles are. I tried hooking it with a pic. No joy. Pulling on it only made it worse. But that white hot cleaning rod did the trick in about 3 seconds.
GI cleaning rods are bullshit.
Use a one piece coated cleaning rod.
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Did you miss the part where it was heated white hot and used to melt through a boresnake?
That may be the only appropriate use of one. 
I’m just making statements in general.
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Fun fact - the hottest man made temp ever is 9,900,000,000,000°F
After I posted I was wondering if there was some hilarious joke that everyone else got (the zip ties went over my head for sure).
I had a similar experience with an Otis Ripcord. I chalked that up to a one time thing, and bought another. I still have some hesitation about using it, but if it ever gets stuck again, I’ll just hit reverse and pull from the chamber end.
The OP’s story smells to me.
Personally, I think that bore snakes make for an adequate substitute for patches used after scrubbing with a brush. But, if you use a snake that has a scrubbing portion that is shorter than your barrel and chamber, then you run the risk of having the pull cord portion snap off when the scrubbing portion is sompletely in the barrel. Do so at your own risk, and don’t use pistol length snakes in rifle barrels.
Never had this happen with an Otis Ripcord.
I’ve broken several boresnakes and don’t use them anymore except for 12 and 20 gauge.
I’m curious if this was one of the Viper style or not. I don’t know if it’s valid or not but I’ve heard they’re more prone to this very problem for some reason.