Can you hurt yourself with a 'brain freeze'?

You know, that intense pain you get above to roof of your mouth if you drink a really cold beverage too fast.

As bad as it hurts, is there potential there for real harm? It’s hard for me to believe that the human body would give such a strong warning if there wasn’t potential for damage.

(And it you have a fix, cure or way to lessen a brain freeze, you can post that as well.)