[JoeyBob]

Our textbook, for instance, says E2 can occur on a tertiary carbon-halide with any base. Water is a base.

Does this mean both E1 and E2 can occur on a tertiary carbon-halide if water is used? How would I know which to choose?

[sharbeldam]

E2 prefers strong bases, I would say that water would prefer E1. But in my experience you always get mixes.

[hollytara]

The original work by Spieth, Ruebsamen and Olson (JACS 1954, v 76 p 6253): they kept t-butyl chloride in water/methanol solutions at temperatures below 30 C and measured rates and products.  They get about 15-20% isobutene and 80-85% t-butanol and MTBE as products with no base - so it looks like E1 dominated even at these low temperatures. 

They did some with a little added base (as the solvolysis proceeds, HCl is given off and then this can catalyze addition of water or methanol back to any alkene products!) to prevent loss of the isobutene to acid catalyzed hydration.  The fraction of elimination was the same. 

Verdict: you need higher concentrations and a stronger base to go to E2.  How strong and how high is another question.  But the classic conditions (>1 M ethoxide in refluxing ethanol) definitely go E2.