I don't think it's a matter of actually believing something is impossible, and being arrogant in stating it. That's oversimplifying matters.

What most people mean when they say something is impossible, is either that they can't muster the necessary will to do it, or aren't willing to make the attempt.

It's like losing weight - we all know what it takes to lose weight, or get fit, we all know we can do it, so when we state it's impossible, what we are saying is that we aren't willing to put in the work. We don't want it bad enough, in other words.

I do HEMA (historical fencing), and from time to time I think to myself "damn, it would be great if I was good enough to enter the big tournaments, but that's impossible". I know it isn't impossible, of course it isn't, it's just a thought. It isn't even that anything is holding me back, it's that I'm not willing to invest the time and effort necessary to achieve it. I don't want it enough, or I want other things more.

In the case of losing weight, it's a matter of enjoying food more than you enjoy being fit - not because people truly believe it's impossible.

But that's just my two cents.