Understanding the contact between solid surfaces is a long standing problem which has a strong impact on the physics of many processes such as adhesion, friction, lubrication and wear. Experimentally, the investigation of solid/solid interfaces remains challenging today, due to the lack of experimental techniques able to provide sub-nanometer scale information on interfaces buried between millimeters of materials. Yet, a strong interest exists improving the modeling of contact mechanics of materials in order to adjust their interface properties (e.g. thermal transport, friction). We show here that the essential features of the residual gap between contacting surfaces can be measured using high energy X-ray synchrotron reflectivity. The presence of this nano-gap is general to the contact of solids. In some special case however, it can be removed when attractive forces take over repulsive contributions, depending on both height and wavelength of asperity distributions (roughness). A criterion for this instability is established in the standard case of van der Waals attractive forces and elastic asperity compression repulsive forces (Hertz model). This collapse instability is confirmed experimentally in the case of silicon direct bonding, using high-energy X-ray synchrotron reflectivity and adhesion energy measurements. The possibility to achieve fully closed interfaces at room temperature opens interesting perspectives to build stronger assemblies with smaller thermal budgets.