Based on a general war:
In theory, the Internet is a decentralized network in which there are multiple routes from computer to computer. Even with widespread destruction - from bomb strikes, sabotage, power outages, whatever, this would remain true. But..
Most Internet bandwidth will come from major PoPs (Point of presences) and their high speed lines, owned by major telcos. So a city may have the vast majority of it's internet traffic going through perhaps 2 or 3 major switch centers. Take these out and most of your bandwidth is gone. Take out the fibres connecting these PoPs and again, you lost most bandwidth. No doubt alternative routes will be found but at a fraction of the bandwidth.
Most of the major sites exist on server farms in data centers. Think Google (including all their services), YouTube, Facebook, Wikipedia, Stack Overflow (the horror!), eCommerce, etc. All based in data canters. In a major war, expect most of these to at least lose power if not physically damaged. So you'd be back to running your own server. Most of what we
If the root DNS system is taken down, then IIRC most of the World Wide Web would also go down.
Most of what we think of as the Internet would probably go down. Network geeks would probably be able to use something based on older pre-Web technologies with much lower bandwidth requirements, but the modern internet (which is far more of a centralized model nowadays) would mostly go down.