As grandpa said not sure which modem, although I don't think it will make a lot of difference. It almost sounds like the cables in some way. I mean lights on and off with different cables. In a regular eletric connection, that would almost tell me some of the cables have bad ends maybe. I am thinking cables.
Yep my first guess is a handfull of bad cables. It only makes sense. So you might need to replace all but the one then.
Or simply bad connectors on the modem itself. Meaning where the cables connect to the modem. If there were a way to tighten the sockets.