In the plot of the Eleven Blue Men, the incubation period is too short for the most common bacterial poisonings, and the main symptoms are diarrhea and vomiting, which be present as a specific response to ailments like gastritis or poisoning, or as a non-specific sequela ranging from brain tumors and elevated intracranial pressure to overexposure to ionizing radiation.
The men do not have the main symptoms of food poisoning: diarrhoea and vomiting. 
