How could Bohr-Bury model predict the $2n^2$ rule without applying the quantum theory? 
In 1921 Bohr and Bury gave the rule that  the maximum number of electrons in any shell is given by $2n^2$. In 1926 the concept of quantum mechanical model was established.
Normally the rule of $2n^2$ rule is explained by the application of magnetic quantum number.
But how did Bohr-Bury predict the in 1921 that the maximum number of electrons is $2n^2$?

My tryout...
I think this hypothesis is in line with radioactivity which was a bubbling topic at that time.
I also thought that the electrons may move as couples, one spin clockwise while the other in anticlockwise direction, thereby having a stablising effect.
Thus the term 2* comes forward.
 A: Bury's paper on the subject is available here.
The argument was based on the structure of the period table i.e. there are:


*

*one row of two elements (H, He)

*two rows of eight elements (Li-Ne and Na-Ar)

*a row of eight elements plus ten transition elements

*etc


It was known from the Bohr model of the hydrogen atom that electrons occupied shells, so the question is how these shells fill up to give the structure observed in the periodic table. That means the first shell must hold 2 electrons and the second shell 8 electrons. Thereafter some handwaving is involved to explain why the third row of the periodic table only holds eight elements not 18, though Bury's argument that some shells fill up before others is pretty good as we now know the 4s orbital fills up before the 3d.
The argument is largely ad hoc, which is fair enough given how little was known about atomic structure at the time. Arguments were based on relating the numbers of electrons to the volume and area of the shells.
A: The $2n^2$ rule complicating the explanation of the periodicity of the atoms in the Periodic table of elements. The subshells are not following in the right sequences. But scientists love to express things with mathematical formulas.

Another sketch for the real arrangement of the shells see here:

A: Bohr's atom does not give any formula for how many electrons occupy a shell.
The number of electrons in an atom come from Morseley's experiments on the third row of the periodic table, which showed that the central charge Z increased by one as one stepped across the table.
One could then deduce that the shells were occupied from inside to out, using two electrons per orbital, and that the s and p layers correspond to the valence electrons.  (Valance is an outside covering).
The two electrons per orbital are explained by the spin-up and spin-down first observed in the zeeman splitting of lines in a magnetic field.  These are explained ultimately as electrons are fermions.
