In reading about Faraday cages, I have found that most "real world" designs incorporate multiple layers of mesh. For example, this article from the National High Magnetic Field Lab discusses several different shielded rooms built of varying layers.
In the MagLab's High B/T Facility at the University of Florida's Microkelvin Laboratory in Gainesville, for example, all electromagnetic waves must be excluded. The magnet there is in a "tempest" quality shielded room, featuring walls made of layers of copper and welded steel that absorb the entire spectrum of EM radiation.
My presumption is that each layer is tuned to or built for a particular range of frequencies. It's not clear to me if the tuning is due to thickness requirements for the layer and therefore a cost factor, or if there are other factors driving the design of each layer.