Nucleonic matter displays a quantum-liquid structure, but in some cases finite nuclei behave like molecules composed of clusters of protons and neutrons. Clustering is a recurrent feature in light nuclei, from beryllium to nickel. Cluster structures are typically observed as excited states close to the corresponding decay threshold; the origin of this phenomenon lies in the effective nuclear interaction, but the detailed mechanism of clustering in nuclei has not yet been fully understood. Here we use the theoretical framework of energy-density functionals, encompassing both cluster and quantum liquid-drop aspects of nuclei, to show that conditions for cluster formation can in part be traced back to the depth of the confining nuclear potential. For the illustrative example of neon-20, we show that the depth of the potential determines the energy spacings between single-nucleon orbitals in deformed nuclei, the localization of the corresponding wavefunctions and, therefore, the degree of nucleonic density clustering. Relativistic functionals, in particular, are characterized by deep single-nucleon potentials. When compared to non-relativistic functionals that yield similar ground-state properties (binding energy, deformation, radii), they predict the occurrence of much more pronounced cluster structures. More generally, clustering is considered as a transitional phenomenon between crystalline and quantum-liquid phases of fermionic systems.