Membrane fusion and fission are key processes underlying, e.g., synaptic transmission. For a quantitative characterization of these processes at the molecular level, understanding the physics of lipid bilayer interactions at close distances is essential. We have observed increased membrane thickness for early membrane fusion intermediates in SNARE-mediated membrane fusion, resembling similar effects observed previously for multilamellar-membrane stacks. To reveal the underlying structural changes and molecular causes, we have carried out all-atom molecular dynamics simulations of double-membrane systems at decreasing mutual distances. Indeed, for distances below 1.5nm, thickening is seen also in the simulations, although to a smaller extent (11%). We also observed a decreased area per lipid and increased lipid chain order parameters, strongly reduced translational and rotational entropy for water molecules between the membranes, as well as markedly slowed and tightly correlated lateral diffusion of the inwards-facing lipid molecules. Particularly for close distances, the dipolar lipid head groups become increasingly tilted, which we identified as the main determinant for the observed thickening.