I'll go along with the general suspicion that it's a problem with the common
wire in the analog cabling. For most analog dimmers, when the common is broken or not making contact, the signal
current for one 'on'
dimmer is more or less able to flow back along along the wires for the 'off' dimmers, but as more and more dimmers are turned on, the available paths are reduced and the electrons become gridlocked, with the result of lights getting darker and darker as more channels are added, just as you are experiencing.
If the problem is minor corrosion on the analog control plugs, simply plugging and unplugging the connectors several times will often self-clean the contacts. Also check to make sure that plugs are fully seated, as simply being loose in the
receptacle can cause an incomplete
circuit.
The second main possibility is a broken
wire inside one or more of the cables. You can either open the
connector shells and look for obvious things like
wire ends floating around loose inside, or use a voltmeter to check for
continuity between the corresponding pins on either end of the cable. Wires do occasionally break mid-cable, but that is rare without visible damage or long-term abuse. If the cable is a
fanout (one big
plug breaking down into several smaller plugs), some research into how they're pinned out might be needed.
More information about what exact model numbers of your equipment, and what sorts of connectors are being used might be helpful for further troubleshooting assistance here.