First, a great number of solids are ionic and metallic substances. Those show a really strong bonding mechanism between elementar particles.
And second, to get your theory to work, you have to take into account the size of molecules. To evaporate or melt anything, you have to break or at least disturb all the bonds each molecule has to its surroundings.
For something as small as H2O, not even hydrogen bonding is strong enough to make it solid. A large molecule like Polyethylene, on the other hand, is only held by weak London forces, but has melting points as high as 145°C, and cannot be evaporated