The Bohr model was extended by Arnold Sommerfeld. Sommerfeld was able to predict all the Hydrogen atom quantum numbers. This theory is referred to today by the name
"Old quantum theory". Please see also, Hazhar Ghaderi's Bachelor thesis.
The basic idea of this theory is that closed classical trajectories (not necessarily circular) lead to discrete quantum numbers. The quantization is obtained through modified condition, now known by the name "Bohr-Sommerfeld condition"
The application of this theory becomes difficult for many electron atoms where exact classical solutions are not available.
This theory was abandoned after the discovery of the Schroedinger equation which gave precise predictions for many electron atoms spectra.
Now, we know that the old quantum theory is only an approximation to the correct quantum theory which is valid in general for large quantum numbers (First order in the WKB quantization).
However, for integrable systems, the Bohr-Sommerfeld is known to be exact or almost exact.
However, this is not the end of the story. In 1980 Sniatycki discovered a deep geometrical meaning of the Bohr-Sommerfeld condition within the theory of geometric quantization.
Sniatycki's work is described briefly in Matthias Blau's lecture notes.
Heuristically Sniatycki's result states that there exists a coordinate system in which the wave functions are concentrated on subspaces of the phase space where the quantum numbers are fixed to their quantized values. These spaces are called "Bohr-Sommerfeld varieties".
Nowdays, This subject is under active research and was applied sussessfully to complicated quantization problems such as the quantization of the moduli space of flat connections.
The orbital wavefunctions of the hydrogen atom, which obey the eigenvalue equation
$$
\left[-\frac{1}{2\mu}\nabla^2-\frac{e^2}{r^2}\right]\psi_{nlm}=E_{nl}\psi_{nlm},
$$
are functions of the separation vector $\mathbf r$ which points from the proton towards the electron. This is a standard trick in the two-body problem and it is done in both the classical and the quantum versions to factor away the motion of the bigger body (which is close to the centre of mass) and leave an effective one-body problem which is easier to treat.
This means that the orbital angular momentum, with total angular momentum number $l$, is in fact the combined angular momentum of the electron and the proton about their centre of mass. In essence, the proton partakes in part of the orbital motion and takes out some of the angular momentum from the electron. (Note, though, that this is classical language and it explicitly does not hold for the hydrogen atom, where the angular momentum of the motion is essentially indivisible.)
This raises an apparent paradox, which is resolved through the fact that the separation vector obeys dynamics through the reduced mass $\mu=1/\left(\tfrac{1}{m_e}+\tfrac{1}{m_p}\right)\approx\left(1-O\left({m_e\over m_p}\right)\right)m_e\lesssim m_e$ of the system, and this is slightly smaller than the electron mass. This slightly enlarges the orbital radius of the electron (since the Bohr radius is inversely proportional to the mass). The velocity stays constant (at $\alpha c$), which means that the angular momentum $L\sim \mu r v$ stays constant as well.
That said, the proton does have spin angular momentum of its own, but this couples weakly to the electronic motion. This coupling is via the same spin-orbit couplings as the electron, but its much higher moment of inertia means that the relevant energies are much smaller, as are the corresponding hyperfine splittings in the spectrum.
Best Answer
The Bohr model wasn't the right theory of all of atomic physics but it described the levels of the Hydrogen atom correctly, due to a mathematical coincidence related to this solvable mathematical problem in the full quantum mechanics.
The integrality conditions of the Bohr model were ad hoc – chosen so that the energy levels as seen in the Hydrogen absorption/emission spectra could be matched – but the most justified starting point to derive them was the Sommerfeld-Wilson quantization condition $$ \int_0^T p_r\cdot dq_r = nh $$ where the integral of $p\,dq$ goes over one orbital period. In this form, it is analogous to the statement in the full quantum mechanics – that replaced the Bohr model in the mid 1920 – that the phase space (space parameterized by the positions $q$ and momenta $p$) is composed of cells whose area (or volume) is equal to multiples of $h=2\pi\hbar$ (or powers of $h$, if there are many coordinates). The orbit encircles an area in the phase space and the area should be quantized.
By a coincidence, this is also pretty much equivalent to the quantization of the angular momentum, $L=n\hbar$. In quantum mechanics, similar conditions hold but for slightly different reasons and the quantization of the angular momentum allows half-integral values, too: $J=n\hbar/2$ where $n$ is integer according to quantum mechanics.
One must separate the explanations in the Bohr model from those in the proper quantum mechanics; they're inequivalent because the models are inequivalent, too. And it doesn't make too much sense to think about the origin of the conditions in the Bohr model because the Bohr model is fundamentally not the right theory as we know today.
In the full quantum mechanics, one may encounter several "quantization" facts with the quantum proportional to $h$ or $\hbar$ or $\hbar/2$. All of them have a quantum origin but the detailed explanation is different for each: the quantization of the angular momentum; the elementary cell of the phase space; the unphysical shifts of the action by a multiple of $h$.