I solved it. Even though we have our phone through the cable company, the lines go outside to the original AT&T junction box.
One of these things:
The cable company just uses it as a place to connect the different lines that go to the phones in the house.
There must have been something funky (like a spider) in that top junction. After much, much trial and erroring, I found that that if I bypassed that thing by just twisting the wires together, everything worked fine.
Once I knew that, I just transferred things down to the junction thingie below that, and now things work fine. That was tough, but I'm glad that there isn't some fault deep in the walls someplace.