This is a deep and fundamental question. It is possible from geometric arguments to prove that in a free-electron model, the only place for conduction band minima and valence band maxima are *at* high symmetry points. Of course, we know that the CBM in Silicon is not at a high symmetry point (it is about 90% of the way towards X on the gamma-X line). This means that the crystal potential of the nearly free-electron gas model is the origin of shifting conduction band minima and valence band maxima off of the high symmetry points. Now, the question exists, is it possible to shift the VBM/CBM off the high-symmetry line? If so, it would be extremely interesting, as such a material would have incredible multiplicity of VBM/CBMs in the Brillouin Zone, which would result in an extraordinarily high Seebeck coefficient (for thermoelectric applications). Alas, to the best of my knowledge, I don't think any such materials exist, but I have no proof if it could or could not exist. The answer either way would probably be a PRL. 

Silicon Electronic Structure: https://nanohub.org/wiki/MaterialScienceSimulationTools/Image:siesta_Bandstructure.jpg

Multiplicity of VBM/CBM for high power thermoelectrics:

http://www.nature.com/nature/journal/v473/n7345/full/nature09996.html

This is a deep and fundamental question. It is possible from geometric arguments to prove that in a free-electron model, the only place for conduction band minima and valence band maxima are *at* high symmetry points. Of course, we know that the CBM in Silicon is not at a high symmetry point (it is about 90% of the way towards X on the gamma-X line). This means that the crystal potential of the nearly free-electron gas model is the origin of shifting conduction band minima and valence band maxima off of the high symmetry points. Now, the question exists, is it possible to shift the VBM/CBM off the high-symmetry line? If so, it would be extremely interesting, as such a material would have incredible multiplicity of VBM/CBMs in the Brillouin Zone, which would result in an extraordinarily high Seebeck coefficient (for thermoelectric applications). Alas, to the best of my knowledge, I don't think any such materials exist, but I have no proof if it could or could not exist. The answer either way would probably be a PRL. 

Silicon Electronic Structure: https://nanohub.org/wiki/MaterialScienceSimulationTools/Image:siesta_Bandstructure.jpg

Multiplicity of VBM/CBM for high power thermoelectrics:

http://www.nature.com/nature/journal/v473/n7345/full/nature09996.html

@Wenhao : The nature link is not functional.

It works for me. But you can try just googling, "Convergence of electronic bands for high performance bulk thermoelectrics"