I had this question asked lately, for the millionth time, so I thought to give it a bit more effort and maybe contribute on the discussion. There are many of us who do both fields, the same way as an artist can also teach math if he is good on it.
To simplify the distinction to extremes, the basic line is:
Designers need a computer, as much as researchers don’t need one.
To be a good designer you need a set of technical skills and tools indispensable for good results. There is a basic technical requirement, under which you cannot function, and you cannot deliver. One must be precise, consequent and methodical to be considered a good designer. The designer is the builder, and as such has to follow the constraint of its universe to the letter. When this is not so, the end result is nonfunctional.
On the other hand, the researcher doesn’t need any precise technical skills. Is good to have a good knowledge of the field you are working in, but no precise technical skills are required. A good researcher has only one indispensable tool, which is the notebook, and psychological and analytical capabilities on the range of a good investigator.
While the designer is constrained by hard math, a researcher is constrained only by Common Sense.
A great example of UX research is Mary Temple Grandin, the American professor of animal science who reinvented the approach of livestock industry on animal treatment and slaughter. Her research was based on her approach of the problem from the animals point of view. The users don’t have to be human for one to be considered a User Experience Researcher, and for that matter there are hundreds of thousand of people in all industries today doing UX research without the title.
So there you have it. Can you do both? Sure you can! But keep in mind that you will not be able to fairly critique you work.