All - thanks for the help!!
The timer, one appliance module, and a light switch is on phase B. The problem circuit, and one appliance module, is on phase A. As a test, I moved the problem breaker to phase B, and had the same results. The results are consistent - when the breaker is off, all appliance modules and light switch work, when the breaker is on, all work inconsistently (sometimes on/off commands work, most often they do not...). I also ran these tests with all 220 breakers tripped (to try to eliminate any cross phase coupling by an appliance) - all results were the same.
Ammeter goes pretty low - I'll run some more tests today with specific ammeter range results.
Not sure if the breaker was on or off when wire was disconnected - I'll retest today. I did test with a new breaker - same results.
Thanks again - this one has me stumped!
I had to do something similar a few years back (but my problem was with Insteon light switched on one circuit not responding to X10 signals at night - which turned out to have been due to my neighbor's noisy CFL bulb).
Here are my thoughts:
1. You tested with a new breaker, and got the same results. That probably rules out the breaker.
2. You moved the circuit to the opposite phase, and the results stayed the same. To me, it doesn't appear to be a phase-bridging issue.
3. Since the only thing left is the wiring on the circuit itself, perhaps there is some issue with the wiring (perhaps something isn't grounded well, or something along those lines). It could also be a transformer, or some other device wired into that circuit, that you just can't seem to find.
You may have to trace out how the circuit runs through your house (box-to-box), and try disconnecting it at each stage, to see if you can find where the problem starts. For example, if it runs like this:
Breaker box
Outlet 1
Outlet 2
Outlet 3
Light Switch
Light Fixture
I'd first disconnect it at the first outlet, and test with only the run from the breaker box to the first outlet connected to the breaker (with the outlet disconnected, and the line capped off for safety).
If it works fine at that point, then re-connect the outlet (with the other side disconnected), and test again.
Continue adding one segment at a time, until it stops working.
This will also give you an opportunity to check for other things connected to that circuit (doorbell transformer, perhaps?), that you weren't aware of.
Do you have any GFCI outlets on that circuit? One that is going bad might also produce noise.
What about illuminated light switches (you know, the ones with the little bulb inside to help you find the switch in the dark)? I suppose something like that could also put noise on the line.
If you can't get it working with ONLY the fist segment in place, you might want to see if you can go around that segment, perhaps using a length of Romex that you run (without fishing it - just for testing) from the breaker to the "outbound" side of that first box.
How far you go with testing this is a matter of how comfortable you are working with this (and from your posts it seems you are pretty comfortable), and how committed you are to fixing the problem.
Let us know if we can help further.