Hammer the wooden dowel down the bore after soaking the snake in oil for a couple days if need be, don't try pushing it by hand.
It doesn't matter if it bunches up, it will still compress and drive out.
I see this often enough and have never not been able to get one out.
The worst are when a guy tries pulling the wrong caliber through.
A 30 through a 22 gets quite stuck.
Throw them away when you start to see any kind of fraying.