Here in the Midwest, I can actually think of times that I've had internet access on my computer and limited or no phone service. For people who do not have cell phones, or do not have wired (not portable) landline phones, losing power means losing phone service. We had a blizzard a few years ago that took our power out but I could still use the internet (until my laptop battery died). We do have corded landline phones, though, so I could call if I'd needed too.
I wonder if you can Skype (or whatever) 911?
@psalm23 I'm pretty sure that Skype doesn't support 911 calls. MAYBE if they use the audio-only, the call might go through (if Skype allows 911 to be called), but then there's no telling how the call would be routed, so there's no telling which 911 service you'd actually reach.
Of course, you really run the same risk when you call 911 from a cell phone: the 911 operator you get might not be anywhere near the place you're in at the time you call.