Engineers: If a good faraway cage is not well grounded would this be the cause of its failure?
For internal protection, the FARADAY cage does not need connecting to earth, it may be isolated and standing on insulators. Earthing the cage will allow nice big sparks to be produced, very dramatic but not necessary.
What failed - how do you know?
A Faraday cage does not have to be grounded at all. (Michael Faraday's original was not.) However, there are reasons why you might want to ground it. The problem I see with your design is that the metallic components - foil layers, screens, etc - are not well and truly bonded together, which is necessary to ensure that the voltage on the cage is exactly the same, at every point on the cage. What you've built may not be good enough for EMP protection. However, if your goal is more in the line of protecting yourself against CIA mind control, what you've built may be quite suitable. EDIT: (After seeing your comments) The layers aren't connecting well to each other, too. The best practical Faraday cages I've seen (WW II - era Army electronics lab) were brass or copper sheet and mesh, with continuous solder connections along every seam. The framing was ordinary lumber, but the metallic cage was continuous within. The only non-soldered joints were around the door, where they used what looked like weatherstripping, made out of springy brass. Further edit: I've also seen one in an MRI facility. (Veterinary hospital; they didn't cover it with gypsum board.) Wall-to-wall-and-ceiling copper sheet, all joints continuously soldered. Double doors to the outside; I don't recall how they were connected (electrically) to the walls. (Probably flexible braided cable, at several points.) Must have cost a fortune to build.