Try chucking the end of it in a variable speed drill and see if it is stiff enough for the entire length of the line to twist. You might get lucky and find that a little rotation is all that is needed to free it. Sounds like it might have gotten snagged against a sharp piece of metal and that metal is biting into it as you pull back on it.

If it rotates the full length, I would feed it in just a little while you very slowly rotate it to get that snagged bit free. That is if my theory is correct.

Can't think of anything else to help.