Why do molecular orbitals in solids merge to bands? Why do molecular orbitals in solids merge to bands?

For example: In silicon every atom is sp3 hybridised, but when I merge two of these orbitals then it yields a bonding and an antibonding MO. When a third atom binds to one of the $\ce{Si}$ atoms in this configuration, then again we should get two MOs, separated by the same energy. This way we we don't get so called bands. How do the bands form?

I have tried wikipedia and other sites but they answer in brief.
Let's explain this with your example of silicon that every atom is sp3 hybridised. when you merge two of these orbitals, then it yields bonding and antibonding MO. But  when a third atom bounds, it yields  3 MO (not four as you mentioned) bonding, antibonding and non bonding MO. The number of MO equals the number of atomic orbitals constituting them, according to LCAO theory.
The energy of the third MO is in between the nonbonding and anti-bonding MOs. When you add a forth, a fifth and so on atoms, you will have more and more MOs very close in energy and you will form a band.
This is a qualitative description about the formation of a band.