This paper examines the ambiguity of noun phrases (NPs) with postmodifying preposition phrases (PPs), such as “the triangle next to the circle below the square.” This ambiguity is attributable in part to the difference in attachment sites for the second PP. If it attaches low, it modifies just the NP that is the object of the first preposition, and the resulting structure is right branching. If it attaches high, it modifies the entire preceding NP, and the resulting structure is left branching. However, each of these structures itself has two distinct interpretations. In the case of the high-attachment structure, these different interpretations have been previously noted; we call them stacking and coordinating. In the stacking interpretation, each PP modifies the entire NP to its left, whereas in the coordinating interpretation, it modifies just the head noun to its left (the first noun in the construction). In the case of the low-attachment structure, only the interpretation corresponding to the coordinating one has been noted; we call it alternating. However, a fourth distinct interpretation is possible, which corresponds to the stacking interpretation of the highattachment structure; we call it stuffing. In the alternating interpretation, each preposition has scope only over the head of the NP that is its complement. In the stuffing interpretation, each preposition has scope over its entire complement. In a pilot study, we found that the interpretations based on low attachment are preferred to those based on high attachment by a 2∶1 ratio. Of the two low-attachment interpretations, alternating is preferred to stuffing by a 20∶1 ratio. However, of the two high-attachment interpretations, coordinating is preferred to stacking by only a 2∶1 ratio. In a second pilot study, we examined the pattern of interpretations of phrases with four PP postmodifiers of an NP, which in principle have 112 distinct interpretation types. Eleven of these types were noted in the experimental materials. We provide a detailed analysis of these types and note that the relative preference of the various interpretations found in the first study is preserved.