Why do n AOs only form n MOs? The textbooks I have read introduce LCAO by considering the H2 molecule. In this example, there are only two possible combinations of the two 1s orbitals (in phase and out of phase). 

When there are more than two valence orbitals, however, a greater amount of combinations arise. A common description of the pi-MOs in butadiene is shown below: 
There are two (and I think only these two, assuming all combining p-orbitals are identical) possible combinations of the p-orbitals which (apparently) do not contribute to the bonding in butadiene. These are shown at the bottom of the illustration above.

Why is it that n AOs only form n MOs, even when more than n LCAO are possible? You would think that all possible combinations would contribute a possible electron energy state to the molecule.
If you have $n$ functions (e.g. AOs) you can make a maximum of $n$ new linearly independent functions (e.g. MOs).  If you try to make $n+1$ MOs, then any one of them can be expressed as a linear combination of the other $n$ MOs.  

The most usual way to enforce linear independence is to enforce orthogonality, i.e. all your MOs have to have zero net overlap with each other. All the "correct" MOs you drew are orthogonal to each other, but the "incorrect" MOs are not orthogonal to any of the "correct" MOs.