Motivated by the duality between elasticity and fracton gauge theory, we study an extension of the gauge group that includes the quadrupole moment. Remarkably, we find that spontaneous breaking of the quadrupole symmetry increases the number of massless excitations. This result appears to challenge the well-established paradigm, according to which gauge fields acquire mass through the Higgs mechanism, and the would-be Goldstone bosons are rendered massive. We refer to this phenomenon as the anti-Higgs mechanism. Furthermore, we demonstrate that the quadrupole gauge theory is dual to an exotic class of elastic systems, which we dub incompressible crystals, and the anti-Higgs mechanism in the dual theory corresponds to a phase transition from the incompressible crystal state to an ordinary (compressible) solid.