I would guess that a 'perfect' Faraday Cage would need to made of a superconducting material. Anything else will have some amount of leakage. All non-superconducting RF shielding materials will have resistance, inductance, and capacitance, which means that it will have a range of frequencies that are attenuated the most, and outside of that range there will be more leakage. It could be very small, but not zero.

So it comes down to practical RF levels for the device. A digital device like a cell phone will have an RF receiver threshold where it becomes impossible to receive valid data, and you will get zero digital communication. An analog device like a walkie-talkie may technically be receiving a super weak signal, but it is inaudible though the rest of the signal noise that is being received. It's technically there, but overwhelmed.

So I guess the answer is that it depends on the construction of the cage, the materials used, the device you are using, and at what point you consider a weak signal to be designated as 'blocked'.

Someone correct me if I'm wrong.