In practice, often not. eg, Disney sends out "cease and desist" to all copyright violations they are aware of (and I criticise this practice, where they knock off creativity in people who aren't making money while applauding it where people charge for using their stuff). People get scared of "cease and desist" orders and so pull their stuff without checking to see if their use is covered by "fair use" first.
So, what do you think Disney would do if the site refused to comply with the cease and desist? Somehow, I don't see Disney filing a lawsuit.