I was told this as well when I first moved into my neighbourhood.

But I found it really difficult to bundle together the cardboard, especially the little pieces of cardboard, paper roll holders, egg cartons, etc. So I've been putting all the little pieces of cardboard into paper bags, and only bundling the larger pieces of cardboard. Everything has always been collected, and there has never been a problem.

I've noticed that others on my street also do something similar now. So maybe the recycling company in my area is more relaxed about this paper bag issue.