I attended the EM wave propagation lectures years ago at my Uni, so my knowledge is rusty, and the following might not be something that is true in practice.
The faraday cage tries to equalize electrical field potential over space. It does so by allowing electrons to move more or less freely, typically inside a metal mesh.
But
a). metals (unless we use superconductors) have electrical resistance
b). electrons themselves have rest masses (i.e. don't accelerate infinitely quickly in response to force).
So, the movement of electrons is not instantaneous, therefore the electrical field cannot be equalized ideally. in effect we achieve attenuation of the signal, and not its complete blockage.
Secondly, the mesh must have some relation to the signal frequency (the wave-length), probably at 1:1 or 1:2 ratio. At some frequencies (terahertz, infra-red, visible, xray) you'll need pretty much a solid box to attenuate those.
And there's obviously tech which you cannot block/attenuate, like neutrinos. Producing and receiving such signals is however non-trivial and requires particle accelerators and large cavities filled with mineral oil and bulky detectors, so we can probably skip this :). Gravity waves belong to the same category.
You can also probably try to send and receive particles, like electrons or alpha particles if you use non-solid meshes.