Paradoxes, in any subject, have the interesting feature of making the subject come alive. They not only make students of the subject examine their theoretical models closely, but also provide insight and directions for exploring the subject further while making them aware of the limitations of the models they use. In this paper we examine several paradoxes that arise in incompressible flow theory with the intent of showing that the frequently used commonsensical arguments, such as, small causes produce small effects, symmetric causes produce symmetric effects, three‐dimensional phenomena have two‐dimensional and one‐dimensional counterparts, steady‐state behaviour can be studied by examining steady‐state equations etc., can actually be very naive arguments in some cases.