We study how a cavity changes the response of hollow microgels with respect to regular ones in overcrowded environments. The structural changes of hollow poly(N-isopropylacrylamide) microgels embedded within a matrix of regular ones are probed by small-angle neutron scattering with contrast variation. The form factors of the microgels at increasing compressions are directly measured. The decrease of the cavity size with increasing concentration shows that the hollow microgels have an alternative way with respect to regular cross-linked ones to respond to the squeezing due to their neighbors. The structural changes under compression are supported by the radial density profiles obtained with computer simulations. The presence of the cavity offers to the polymer network the possibility to expand toward the center of the microgels in response to the overcrowded environment. Furthermore, upon increasing compression, a two step transition occurs: First the microgels are compressed but the internal structure is unchanged; then, further compression causes the fuzzy shell to collapse completely and reduce the size of the cavity. Computer simulations also allow studying higher compression degrees than in the experiments leading to the microgel's faceting.