The popularity of an opinion in one's direct circles is not necessarily a good indicator of its popularity in one's entire community. Network structures make local information about global properties of the group potentially inaccurate, and the way a social network is wired constrains what kind of information distortion can actually occur. In this paper, we discuss which classes of networks allow for a large enough proportion of the population to get a wrong enough impression about the overall distribution of opinions. We start by focusing on the 'majority illusion', the case where one sees a majority opinion in one's direct circles that differs from the global majority. We show that no network structure can guarantee that most agents see the correct majority. We then perform computational experiments to study the likelihood of majority illusions in different classes of networks. Finally, we generalize to other types of illusions.