In the quest for materials addressing the grand challenges of the future, there is a critical need for a broad understanding of their electronic structures because the knowledge of the electronic structure of a given solid allows us to recognize its structural preferences and to rationalize its properties. As previous research on quaternary chalcogenides containing active metals (a group-I- or -II-element), early transition-metals, and late transition-metals indicated that such materials could pose as alluring systems in the developments of thermoelectrics, our impetus was stimulated to probe the suitability of tellurides belonging to the prolific A3R4Cu5Te10-family. In doing so, we first used quantum-chemical techniques to explore the electronic and vibrational properties of representatives crystallizing with different A3R4Cu5Te10 structure types. The outcome of these explorations indicated that the aspects that control the formation of a given type of A3R4Cu5Te10 structure are rather subtle so that transitions between different types of A3R4Cu5Te10 structures could be induced by manipulating the ambient conditions. To probe this prediction, we explored the thermal behavior for the example of one quaternary telluride, that is, Rb3Er4Cu5Te10, and thereby identified a new type of A3R4Cu5Te10 structure. Because understanding the structural features of the A3R4Cu5Te10 family plays an important role in the analyses of the aforementioned explorations, we also present an overview about the structural features and the members of this class of quaternary tellurides. In this connection, we also provide a structural report of four tellurides, that is, K3Tb4Cu5Te10 and Rb3R4Cu5Te10 (R = Tb, Dy, Ho), which have been obtained from high-temperature solid-state reactions for the very first time.