Multiquark states have been advocated to explain recent experimental data in the heavy-light sector, and there are already speculations about multiquarks containing only heavy quarks and antiquarks. With a rigorous treatment of the four-body problem in current quark models, full-charm and full-beauty tetraquarks are found to be unbound. Thus their stability should rely on more subtle effects that are not included in the simple picture of constituent quarks. The case of might be more favorable if the naive color-additive model of confinement is replaced by a string-inspired interaction.