The width of the walls scales relative to the wavelength of the frequency you're trying to stop. So it doesn't take much to block all useful signals.
You should realize that aluminum foil blocks visible light signals.
Yup, I mentioned the solid box earlier - a non-mesh metal container.
Reading through some more material, it seems that there are ways around it - one can use a static electric field to send signals, e.g. switching it on/off or changing its intensity. I suppose the differing el. charge of the opposing walls of such a cage/box would create en electrical field gradient within such box, which is easy to detect, if one's goal is sending some low-bitrate signals.
It's detectable (i.e. the fact of using it), but the communication scheme itself can be encrypted or encoded, so plausible deniability can be employed.