It seems that WKB is applicable for a given $E$ if and only if $\hbar$ is sufficiently small. Or in other words, WKB is applicable if and only if the quantum number is large enough.
Is this understanding right?
I would take the exactness of WKB for the harmonic oscillator as purely accidental.
Comments to the question (v3):
Yes, given a Hamiltonian $H$, the semiclassical WKB method can give a qualitative but not a quantitative prediction for the ground state energy $E_0$.
The WKB prediction (incl. the metaplectic correction/Maslov index) for the harmonic oscillator (HO) happens to be exact due to a hidden supersymmetry, cf. this Phys.SE post.
For an example where WKB is not exact, see e.g. my Phys.SE answer here.
The WKB approximation can be derived using an expansion in powers of $\hbar$. However, that doesn't imply that it can only be used if some quantum number is big. For example, a classic application of the WKB approximation is to alpha decay, which typically occurs from the ground state.
@DanielSank, for your question about "expanding in h_bar", please see the following explanation from http://www.tcm.phy.cam.ac.uk/~bds10/aqp/handout_approx.pdf.
