Time Warner’s phone does go down during power failures. I switched to ATT Uverse, their phone is battery backed up.
AT&T’s land line phones used to be battery backed up. In some areas not anymore. Had an ice storm last winter and the power was out for 3 days. Only cell phone worked. When the power came back on, AT&T DSL worked but the land line was still dead for days. Land lines in a large area was out. Seems like the reverse should have happened. These are phone lines that stayed on during and after Hurricane Hugo.