Is it possible for a crystal to have different structures at different temperatures? For instance, suppose it is a 50-50 alloy of two metals that is BCC at room temperature $T_0$. If I raise (or lower) the temperature, is it possible for the bonds in the crystal to rearrange and form a new structure (say, FCC) that is more energetically favourable at the new $T$?
I saw a question on an engineering forum (here) that seemed to suggest the only thing that affected the structure was the % composition of the alloy. Could temperature play a role, too?
 A: Yes, it is very possible.  Even water goes through such different structures

Two lines in particular from the wikipedia article on Ice:


*

*Ice II    A rhombohedral crystalline form with highly ordered structure. Formed from ice Ih by compressing it at temperature of 190–210 K. When heated, it undergoes transformation to ice III.

*Ice III   A tetragonal crystalline ice, formed by cooling water down to 250 K at 300 MPa. Least dense of the high-pressure phases. Denser than water.

A: No alloy is required. Plutonium is an example:

A: 
The microstructure of an alloy depends on such variables as the
  alloying elements present, their concentrations, and the heat
  treatment of the alloy (i.e., the temperature, the heating time at
  temperature, and the rate of cooling to room temperature).
  -Materials Science and Engineering: An Introduction 9th, Wiley, Calister, Rethwisch

Look up phase diagrams for different alloys
Here is a phase diagram for iron and carbon (steel). Depending on the composition of the alloy and the temperature it is at different crystal structures will form and multiple phases can be present at the same time. $\alpha$ is ferrite and has BCC structure, austenite ($\gamma$) has FCC.

A: There is certainly far more to metallurgy than % composition of the alloy. Devin Crossman's answer hints a little at the processes involved. Hardness is strongly affected by heat treatments, for example, though this does not have much to do directly with the subject of this question https://en.wikipedia.org/wiki/Hardening_(metallurgy).
A classic (and very relevant) example of a phase change in metallurgy occurs with tin. Tin changes from a tetragonal metallic structure (density 7.365 g/cm3) to a diamond type nonmetallic structure (density 5.769 g/cm3) at temperatures below 13.2C.
The transformation is said to be "autocatalytic" (though I think it may be more a case of crystal nucleation) and is referred to as "tin pest." Tin objects will become brittle and simply fall apart at low enough temperatures.
https://en.wikipedia.org/wiki/Tin_pest
https://www.youtube.com/watch?v=FUoVEmHuykM
