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I would ask about this situation: what is there in common between the two consoles? since they both display the same problem.
- the Multi OUT connector on the Nintendo64 console ?
- the AV cable otherwise? but you've tested it with Gamecube.
- The TV.... make sure it is on the right channel before you hook up the game console. Also make sure that the SOURCE selection is correct, for the RCA Video and RCA Audio (red, white, and yellow) Input. If the TV is not set to the correct Source, it will say No Signal (unless the source is producing some sort of signal it can read and turn into a picture and sound.)
Also, of course, blow into the game cartridge before inserting it. Examine all game cartridges for dirt or dinginess on the metal contacts/terminals on them, before inserting them, and if it looks necessary, clean those metal contacts off with cotton swabs moistened in rubbing alcohol, allowing 5 minutes or more for drying before trying the cartridge again.