We consider thermodynamically V-representable one-matrices, i.e., one-particle density matrices that are obtained by reducing the Gibbs grand canonical density matrix of a quantum mechanical many-particle system subject to a suitable external potential v, and show them to obey an inequality lower bounding their eigenvalues in terms of those of the one-particle kinetic energy operator. The result imposes a severe constraint on the asymptotic behavior of the eigenvalues of any one-matrix to be V-representable. For noninteracting particles, the corresponding upper bound is also proven, implying that a one-matrix can be interactionlessly V-representable for at most one temperature. We expect the upper bound to be valid more generally, as is illustrated by a model of coupled harmonic oscillators where the V-representable one-matrices can be explicitly calculated, and discuss its implications for certain aspects of density-matrix functional theory.