Down For Everyone Or Just Me? Down For Everyone Or Just Me? won't fix the problem of not being able to access a site, but it will at least let you know if your computer or network is the cause of the problem.
Along the same lines of determining if a site is down or not, the incognito mode in Chrome and privacy mode in Firefox can be useful when you want to see or demonstrate how a site will look to your students when they visit it for the first time.
There are two other occasions on which I use incognito mode in Chrome. First, I use incognito mode when I'm demonstrating how to use a web app or website that requires my username and password. Second, I use incognito mode when I want to show people what a shared or published Google Document looks like when you're not signed into a Google Account. In both cases opening a new browser in incognito mode saves me the hassle of signing out of an account when I know that I'm going to be back into it in a few minutes. Incognito mode thinks that I'm a new user and doesn't have my saved usernames and passwords so I'm able to show sites and documents as they appear when I'm not signed into them.