Only when anomalies stack up and increase to make the worldview unsustainable.

My experience shows that for most people the new facts must reach a point where the cognitive dissonance becomes almost physically painful before they will change their mind about anything.

Then they will probably hate you for changing their mind, because they decide that you are the source of the discomfort, which is false, because you don't control the Truth of the world.

If parts of their worldview are connected to their identity, then they would probably rather die than change. For example, lefties who are deprogrammed enough to see that the cause they were advancing causes almost unlimited misery and death. To discover that you have been so badly wrong is intensely painful should cause deep reflection.

I am strange in that I have cultivated a willingness to be shown how I am wrong, specifically, so I can be less wrong in the future. It is a deeply unnatural mode of thinking and requires rigorous application to develop. I am still struggling with it. Being a devotee to the capital T truth of the world does not make one a nice person, nor is it advantageous for making friends or cultivating social relationships. I'd rather be right than happy. I won't lie, even to myself.