In a series of publications, we have proposed a foundational system of ontological categories which has been used to evaluate and improve the quality of conceptual modeling languages and models. In this article, we continue this work by employing theories from Formal Ontology, Cognitive Psychology and Philosophical Logic to systematically investigate some important modal aspects of the ontological categories represented in structural conceptual models. In particular, we focus on Object Types and Part-Whole Relations, formally characterizing some modal properties that motivate the proposal of a number of distinctions within these categories. In addition, we show how two types of modality known in philosophical logic (de re/de dicto modality) can be used to address some subtle issues that appear in conceptual diagrams when different sorts of object types and part-whole relations are combined.