Why can't bicyclic structures undergo E2 elimination?
Elimination via a E2 mechanism cannot take place at the bridgehead carbon of a bicyclic compound due to the lack of the required trans geometry between the leaving group at the bridgehead carbon and any of the six β-hydrogens. 

On the other hand, as you have correctly observed, a E1 mechanism has to be ruled out as well due to the strain at the bridgehead carbocation. In fact bicyclic structures prevent the tertiary bridgehead carbon becoming planar, which implies that is very high in energy because the nonplanar structure forces the cation to be an empty "sp3 orbital" instead of an empty "p orbital". This observation was made for the first time by Bredt, therefore is generally known as "Bredt's rule".