No, the Bohr-Sommerfeld model is conceptually a classical toy model (which has only been fudged to imply some selected quantization features similar to quantum mechanics) so it is inequivalent to the proper quantum mechanical description or any approximation of it. The agreement of the Bohr-Sommerfeld model with the right quantum mechanical results is a coincidence, a special feature of the hydrogen atom.
The only limit in which quantum mechanics reduces to the "Bohr-Sommerfeld physics" is the limit of long distances and high momenta in which $\Delta p\cdot \Delta x\gg \hbar$ and in which both quantum mechanics and the Bohr-Sommerfeld model reduce to classical physics without any Bohr-like quantization restrictions. But this limit is clearly not relevant for the description of low-lying states of the hydrogen atom.
Well, some proper interpretations of the Bohr quantization rules also emerge in the semiclassical (next to leading) WKB approximation of quantum mechanics. But one must be careful about the interpretation and various shifts and subtleties. For example, $\int p\,dq$ over the phase space contours is a multiple of $2\pi\hbar$. In Bohr's old picture, this statement applied to allowed closed trajectories of particles. In quantum mechanics, however, it applies to boundaries of phase space regions corresponding to $N$ microstates. The interpretations are slightly different.
In QM, there are typically no closed trajectories as the initial localized wave packets spread, and in Feynman's approach, one sums over all classical trajectories, whether they obey classical equations of motion and Bohr's quantization conditions or not.
Bohr Model of the Hydrogen Atom - Energy Levels of the Hydrogen Atom
All the criticism of my answer below is completely invalid.