Yes. "The internet" means all internet access, whether through cable, DSL, phone, or even dial-up. If you only have one of these, there is no way for you to know that "the internet" is down. All you can say is that your particular provider can't provide internet.

BTW, there are other parts of "the internet" that can be accessed even if DNS goes down and keeps you from getting on the World Wide Web. Most people only know WWW. They don't know about Usenet, Telnet, Gopher, or the other old parts of the internet. I used those other parts before the WWW started.