Aqueous suspensions of aggregated silica particles have been dewatered to the point where the colloidal aggregates connect to each other and build a macroscopic network. These wet cakes have been compressed through the application of osmotic pressure. Some cakes offer a strong resistance to osmotic pressure and remain at a low volume fraction of solids; other cakes yield at low applied pressures, achieving nearly complete solid/liquid separation. We used small angle neutron scattering and transmission electron microscopy to determine the processes by which the particles move and reorganize during cake collapse. We found that these restructuring processes follow a general course composed of three stages: (1) at all scales, voids are compressed, with large voids compressed more extensively than smaller ones; the local order remains unchanged; (2) all voids with diameters in the range of 2-20 particle diameters collapse, and a few dense regions (lumps) are formed; and (3) the dense lumps build a rigid skeleton that resists further compression. Depending on the nature of interparticle bonds, some cakes jump spontaneously into stage 3 while others remain stuck in stage 1. To elucidate the relation between bond strength and compression resistance, we have constructed a numerical model of the colloidal network. In this model, particles interact through noncentral forces that are produced by springs attached to their surfaces. Networks made of bonds that break upon stretching evolve through a plastic deformation that reproduces the three stages of restructuring evidenced by the experiments. Networks made of bonds that are fragile jump into stage 3. Networks made of bonds that can be stretched without breaking evolve through elastic compression and restructure only according to stage 1.