In supersymmetric (SUSY) theories with extra dimensions the visible energy in sparticle decays can be significantly reduced and its energy distribution broadened, thus significantly weakening the present collider limits on SUSY. The mechanism applies when the lightest supersymmetric particle (LSP) is a bulk state-- e.g. a bulk modulino, axino, or gravitino-- the size of the extra dimensions larger than ~10^{-14} cm, and for a broad variety of visible sparticle spectra. In such cases the lightest ordinary supersymmetric particle (LOSP), necessarily a brane-localised state, decays to the Kaluza-Klein (KK) discretuum of the LSP. This dynamically realises the compression mechanism for hiding SUSY as decays into the more numerous heavier KK LSP states are favored. We find LHC limits on right-handed slepton LOSPs evaporate, while LHC limits on stop LOSPs weaken to ~350-410 GeV compared to ~700 GeV for a stop decaying to a massless LSP. Similarly, for the searches we consider, present limits on direct production of degenerate first and second generation squarks drop to ~450 GeV compared to ~800 GeV for a squark decaying to a massless LSP. Auto-concealment typically works for a fundamental gravitational scale of M_*~10-100 TeV, a scale sufficiently high that traditional searches for signatures of extra dimensions are mostly avoided. If superpartners are discovered, their prompt, displaced, or stopped decays can also provide new search opportunities for extra dimensions with the potential to reach M_*~10^9 GeV. This mechanism applies more generally than just SUSY theories, pertainin Show Partial Abstract