It would not be the first time that a product had a bad ground
connection, and that creating other ground paths (galvanic or
capacitively coupled) restored operation.
Some products depend on screw connections (e.g. PCB to case) or
threaded connections to work. One loose screw or nut could compromise
the grounding and signal flow.
Or, there might be e.g. a cracked solder joint, where the cable
connection is attached to the PCB... this can be a "high physical
stress" location if the cable flexes. Check the jack-to-PCB
mounting... if you see any solder joints which are cracked or look
dubious, reheat/resolder them.
Another possibility is that you have a bad cable... its shield might
be making intermittent contact at one end or the other. Having the
scope hooked up and turned on, could provide an alternate ground path
through the building mains (your building's cable shield should be
bonded to the mains ground, where the cable enters the building). Try
a different "known good" cable... and if possible, a different jack on
your cable wiring. Also, instead of hooking up the 'scope, try a
simple wire connected between the outside threads of the cable
connector, and a known-good grounding point in your house... see if
this restores operation.
I really appreciate the help. I decided I should remove the splitter that
separates the internet and TV signals. I hadn't remembered, but the splitter was
a "2-way splitter amplifier, which I had used to get a strong enough signal for
my computer TV tuners to get good digital signals.
With the cable plugged directly into the modem, it works great. With a splitter
for it and the TV, (5-1000MHz, amplified or not), it can't drive the modem. So,
it appears I have a signal strength, quality, and/or a splitter problem, or a
weak modem. It is good to know that it seems to work good with the direct cable
connection, so maybe this is a solvable problem. More experimentation tomorrow.