So far, the well known two branches of real discrete spectrum of complex PT-symmetric Scarf-II  potential are kept isolated. Here, we suggest that these two need to be brought together as doublets: Eⁿ_±(λ) with n=0, 1,2.... Then if strength (λ) of the imaginary part of the potential is varied smoothly some pairs of real eigenvalue curves can intersect and cross each other at λ =λ_∗ ; this is unlike one-dimensional Hermitian potentials. However, we show that the corresponding eigenstates at λ =λ∗ are identical or linearly dependent denying degeneracy in one dimension, once again. Other pairs of eigenvalue curves coalesce to complex-conjugate pairs completing the scenario of spontaneous breaking of PT-symmetry at λ = λ_c. To re-emphasize, sharply at λ = λ_∗ and λ_c, two real eigenvalues coincide, nevertheless their corresponding eigenfunctions become identical or linearly dependent and the Hamiltonian looses diagonalizability.