I suspect that it works this way (using telco descriptions)
[ Home ] --- [ Plant ] --- [ Central Office ]
Plant is everything from your home back to the CO, so cable, repeaters, routers, pole mounted equipment, huts, etc. back to the CO.
If you lose power to your home, your battery backup or generator can supply power and you'll be fine as long as that lasts.
If you lose power on the street any Plant powered by that power will only last as long as it's backup power lasts (I have Comcast whose battery doesn't look very large and it's working fiber which is power hungry).
After Sandy Comcast and Verizon had generators on the equipment for a few weeks.
Power loss at the CO (Central Office) is usually handled by batteries (less so now) and generators.
The Plain Old Telephone Service has federal requirements for backup power and that should continue to operate under many of the worst conditions. While about 1 in 3 calls failed in my area, we had tone, we received calls and could make calls after a few minutes of trying. FIOS and Xfinity were dead until the backup was established. This was a very nasty disaster.
BTW, I'm keeping my POTS line for as long as I can. It may be 100+ year old tech but it's made to work in the worst.
Don't take any of this as a criticism of the Telco or cable companies. They worked very hard in our area to restore all service.