The Mystery: A Florida phone number in the 941 area code.... I was able to dial it with my TW phone a month ago, but this week I get a recorded voice, "The number cannot be connected[?] as dialed." I called TW, they said a tech would call me back in 48 to 72 hours, and indeed, one did. He can't reach the number on his TW test phone, or on a Verizon line, but he had me try it on my AT&T phone (aka, the iPhone) and indeed it went through then (although it still won't on my TW line).
The phone system is way more complicated than I thought if this is happening.
