The questions is mostly in the title, but might expose another of my misunderstanding of the band structure of solids and how that leads to metals and insulators.
If we have a solid, and the fermi energy lies at the top of one of the bands, it will be an insulator, because there are no thermal fluctuations large enough to allow the electrons to move up to the next available state.
On the other hand, if the fermi level is in the middle of one of the bands, the solid will be a metal. If we apply a field, the electron can easily move up to a higher state within the band, and we will have conduction.
Here's my question: doesn't it take thermal fluctuations to "smear" the energy levels within a band? A band isn't really a continuous $\epsilon(k)$ spectrum; it's a series of discrete $\epsilon_k$ values. Does that not mean that at truly zero temperature, we would have the same (insulating) situation as above?