We discuss some implications of the screening mechanism exhibited by massive gravity theories such as the Dvali-Gabadadze-Porrati model. The first one is a new scheme to break Lorentz invariance that results in Minkowski space even when gravity is included. This considerably alleviates the bounds on Lorentz violation, which under the assumption of ordinary gravity are quite stringent. One previously known way to break Lorentz invariance without affecting the background is by the condensation of a tachyonic tensor field, which is related to ghost-condensation. We show that another possibility, realized in the DGP model, is by the presence of sources. A simple example is given by a perfect fluid, which sets up a preferred frame without distorting the spacetime. Similarly, low tension domain walls in DGP become 'stealth': they do not inflate and generate no gravitational effects on a 4D observer, even though they break Lorentz and translational symmetries. This also means that domain walls are probes that enable one to distinguish large-distance modified gravity from general relativity even at short distances. As a byproduct, we obtain exact solutions for super-massive codimension-2 branes.