An MCM includes a two-dimensional array of facing chips, including island chips and bridge chips that communicate with each other using overlapping connectors. In order to maintain the relative vertical spacing of these connectors, compressible structures are in cavities in a substrate, which house the bridge chips, provide a compressive force on back surfaces of the bridge chips. These compressible structures include a compliant material with shape and volume compression. In this way, the MCM may ensure that facing surfaces of the island chips and the bridge chips, as well as connectors on these surfaces, are approximately coplanar without bending the bridge chips.