Strictly speaking, if there are no holes worth the name then it is not a cage, it is an enclosure. The whole point of a Faraday cage is to offer as much as possible of the shielding provided by an enclosure, while also allowing you to access the interior.
A characteristic of conductive enclosures is that any incident EM wave sets up eddy currents in the surface. These eddy currents in turn set up magnetic (B) fields which tend to cancel out (or reflect) the incoming B-field. A resistive material will attenuate the eddy currents, reducing the level of cancellation and allowing some of the B-field to leak through. A magnetic material will bind in this B-field better than a non-magnetic one.
Mild steel is slightly resistive and is magnetic, so is quite a good magnetic shield. Aluminium is more conductive, so the eddy currents are stronger and and the resultant B-field weaker. But it is also non-magnetic, so the attenuation of the B-field is less and more may well end up getting through.
I can vouch from my professional experience that interference still gets through both materials, having designed and EMI tested RF power enclosures made from both. (Silver-plated copper has high surface conductivity, but it is expensive and hence impractical for most applications. And it is non-magnetic, so the residual field still gets through). The trick is to ensure the residual interference is harmless, if necessary by hardening your circuit as required.
In the case of the modern ICT (Information and Communications Technology) you ask about, the signal levels penetrating a half-decent enclosure are so pathetic that there is not going to be a problem unless your circuit is a SQUID (superconducting quantum interference device) or similar super-sensitive toy.
Now, a cheaply-made Faraday cage full of unintended gaps and badly-bonded pieces, that is a different threat level altogether!