(From an earlier post of mine.) Member NoTime has mentioned the same thing.
In the ideal case, the Faraday cage should be an unbroken hollow conducting shell. Then there cannot be any electrical field inside due to outside fields. In practice, it is made of fine mesh, and connected to the ground, that is earthed.
A heavy duty properly grounded cage will prevent lightning from passing through. But, theoretically, it's possible to imagine an exceptionally powerful bolt of lightning entering it, because it's a mesh, not an unbroken hollow shell.