  The smallness of the 1-3 lepton mixing angle $\theta_{13}$ and of the
neutrino mass-squared-difference ratio $\Delta m^2_{12}/\Delta m^2_{23}$ can be
understood as the departure from a common limit where they both vanish. We
discuss in general the conditions for realizing the mass degeneracy of a pair
of neutrinos and show that the vanishing of a CP violating phase is needed. We
find that the discrete quaternion group Q of eight elements is the simplest
family symmetry which correlates the smallness of $\Delta m^2_{12}$ to the
value of $\theta_{13}$. In such a model we predict $0.12\lesssim
\sin\theta_{13} \lesssim 0.2$ if the ordering of the neutrino mass spectrum is
normal, and $\sin\theta_{13}\lesssim 0.12$ if it is inverted.
