Yes, those obese mice are called Agouti mice, and the concept overall is called epigenetics. The ON/OFF state of genes can be changed through diet, nurture, and more. For example, if you do not eat very nutritious food you might not have enough methyl groups to methylate (turn OFF) some of the genes that are currently ON but should naturally switch.

http://www.nature.com/scitable/topicpage/obesity-epigenetics... http://en.wikipedia.org/wiki/Epigenetics