Fair use isn't a blanket excuse to do whatever you want. The key component is whether your use is transformative. If you copy something to criticize it, or make something that doesn't compete with the original, it's likely fair use. If you copy something just to have it, or build upon it, it's not. This is a new issue, but copying someone's tweets, just to create your own library of tweets is not really transforming anything, I don't see how this is fair use.