Can I cite email communication in my thesis/paper? In e-mail correspondence, the author of an competing paper has admitted a shortcoming of their method, roughly: "Yes, our method X does not work when used with Y." This is not obvious from their paper and is quite a big point from my perspective.

Can I cite this e-mail correspondence in my master thesis/paper (is there a difference?) and use as motivation for my work? Is this impolite?
If I just say, X does not work with Y without citation, I feel reviewers will not believe me/say I did not apply the method accordingly.
The exact manner of how you should cite the communication may vary from style to style (often the date is included), but it's entirely reasonable to cite such a communication in a paper or thesis. Typically this is referred to as a Personal Communication. Obviously, wherever possible, you should look for information in published sources, but where this is not possible citing such a personal communication will suffice.

From memory, I think there might be one in my own thesis. A specific detail of an experimental method used in a published article was not discussed, but I thought it relevant to my argument so I contacted the author to get the information I needed.

Edit: Typically personal communications should not be included in the reference list as they are not evidence that can be readily recovered by anybody reading your work. This is made explicit in several prominent styles (e.g., Harvard, Chicago, APA).