My mother had AT&T before switching to cable for internet. She was having speed and connection problems for months, but all they would do is send out a tech, who would say the line was fine, and then they'd send her a new modem. After three, I came and looked at her lines from the modem to the demarc, the grey box on the outside of your house where the outside line meets the inside line. It's a two chamber box, with one secured with just a screw, and the other secured with one of those "secure" hex screws (most times, a small enough screwdriver will turn it, but line techs are usually in too much of a hurry to tighten them properly). Since she has a full, unfinished basement, I could visually check the lines and connectors from end to end.
Even after cleaning up contacts, she still had trouble, though not as severe. AT&T sent out another tech, he said the line was clean, and they sent her another modem. Now, the "customer side" of the demarc has a funny little connector that has multiple terminals and an RJ11 jack, for testing. I took one of the (now many) spare modems out with enough cords to get it to power on and connect to the central office equipment. Look in the modem interface, and the speed's still not right, and, fortunately, the connection dropped a couple of times, or I'd've had to leave it out overnight. So, I open the company side, disconnect the leads from the connector and clip on a jack that I kept with alligator clip leads on it. Plug the modem into
that, and it was solid!
Anything "inside" the demarc belongs to the customer, meaning the wiring from the demarc to the jacks, but anything actually inside the demarc belongs to the incumbent carrier. Their testing method assumes that the demarc is a black box that always works, to the point where AT&T and QWest (and I would assume Century Link) outside techs don't even carry them on their trucks.
Man! Sorry about that. TL;DR: Yup, they sure do!