V- (the helpful person) listened to my tale of woe, said maybe it was indeed the outside part (that they still service) and she would talk to Customer Service and get back to me. Which she did. Customer Service told *her* it was probably an inside jack, but V- said she could send someone out to look, since the outside box was so old and all.
Today we had a cheerful, obviously competent repairman show up, who looked at our outside box and said it was old but they were usually good--only when he dug into it, he found corroded this and that and the other. It's definitely over 30 years old. I think it was on the house long before we bought this house. Outcome--we have a new outside box, that has the little jack thing where you can test your own line to see if it's their line or your internal line that's a problem. So now I feel more confident about ordering in the DSL and having real! live! faster-than-this! internet connection. Which, since I need to email off a humongous book manuscript, would be a very nice thing to have.
So in the morning (I had something urgent to do as soon as the repairman left and didn't get done until after business hours) I will call the phone company back and order DSL.
It occurs to me that Customer Service's job is to keep people from actually getting service (for instance, repairs), but the repair service people, once you contact them, are pleasant, helpful, and just generally good folks. The Customer Service people are probably good folks too, out of the office, but the times I've talked to them (when trying to get this "it's noisy or totally out when it rains" situation fixed) they've acted reluctant to move my problem up the line to Repair Service. I keep thinking of George D-, who's in maintenance and repair for a different telco, and is such a rock-solid decent hard-working guy. All the power company and phone company repair people I've ever met, lifelong, have been that way.