Thanks for all the replies. In answer to some questions:
What kind of line is it?
Not sure. It's copper to the demarc at each end, but beyond that I don't know. The buildings are in different exchanges, but I don't know if they might still be located at the same central office. It's supposed to be a dedicated dry pair, but we're pretty sure it isn't. With the tip and ring open at each end, there is about 100 ohms at each end. If you short either end, the opposite still reads 100 ohms. Not looking good for copper end-to-end, although according to my techs you can push a tone from one end to the other. I don't believe it's a T1 line either. There's no need for that bandwidth. The modems operate at 19.2K baud.
Won't Verizon test?
Sure they will, and have several times. According to them, all is unicorns and rainbows. I've never seen a report so I don't even know what they are claiming, exactly.
Routers, network packet analyzers, etc.
This is a strictly analog set up. Tip and ring connected at each end and that's it. The short haul to the computer at each end is 20 - 50 feet.
Next Steps
I've spoken to one of Siemens top trouble shooters, and he said he's seen this sort of thing before. In all likelihood the signals are going digital at some point, which means (if I understand correctly) multiplexing and switches. Now for an important bit of info. The modems are set up to adjust their speed if they detect connection issues. So if the switch gets jammed, they'll drop the connection speed. When traffic lightens up, the modems don't always come back up properly, and this is where zbang's observation regarding slip sync may come into play, especially if the modems are driven down to 1,200 baud by an overstuffed switch. The Siemens engineer said what he has done in the past is tweaked the modems to connect at 9600 baud, period. This has solved the problem for at least one customer in Union county that has a massive number of leased line connections to their central site. Any time they've had an issue after that it's been traced to a Verizon issue.
So Tuesday we have a party planned for the customer's sites. We have a contractor who is very familiar with this gig and has done this sort of work for the Siemens branch. He'll bring in a couple Siemens panels with the tweaked modems and check if that clears the issue. For now we wait with bated breath.