A simple proof that no subset of the plane that meets every line in precisely two points is an Fσ-set in the plane is presented. It was claimed that this result can be generalized for sets that meet every line in either one point or two points. No proof of this assertion is known, however. The main results in this paper form a partial answer to the question of whether the claim is valid. In fact, it is shown that a set that meets every line in the plane in at least one but at most two points must be zero-dimensional, and that if it is σ-compact then it must be a nowhere dense Gδ-set in the plane. Generalizations for similar sets in higher-dimensional Euclidean spaces are also presented.