A study at Columbia University Medical Center, as reported in Science Daily, sheds light on why some people that don't have celiac disease notice symptoms upon ingesting wheat (not only gastrointestinal symptoms, but systemic symptoms such as fatigue and brain fog). It appears that there is evidence of a "weakened intestinal barrier". In the study, people also noted improvement in their symptoms upon excluding wheat from their diet. The researchers next plan to investigate why the intestinal barrier becomes weakened in the first place. I would imagine suspects to be antibiotics, infections, steroid medications, antacids, stress, etc.
Columbia University Medical Center. "Biological explanation for wheat sensitivity found: Weakened intestinal barrier, systemic immune activation may explain symptoms in people without celiac disease." ScienceDaily. ScienceDaily, 26 July 2016. <www.sciencedaily.com/releases/2016/07/160726123632.htm>.