Experimental ways of studying the structure of atomic orbitals As far as I understand, the notion of atomic orbital was introduced by Bohr in 1913 in terms of his semi-classical model of atom. Bohr assumed that each electron in the atom has a certain energy and angular momentum (from the modern point of view of the Schrödinger equation this is true only approximately). Then an atomic orbital is defined to be the group of electrons of given energy and angular momentum. (The modern QM definition is the same but energy and angular momentum of each electron are defined approximately). As far as I understand, this notion helped to explain chemical properties of atoms reflected in the periodic table.
Are there other experimental ways to study the structure of orbitals, e.g. to count the number of electron in a given orbital?
UPDATE: The definition of atomic orbital in the post is not correct. This is due to difference between English and Russian terminology and my incorrect translation.
 A: The brief answer is that, yes, the structure of orbitals can be experimentally studied.
However, the meaning of such an answer is quite different from the meaning emerging from the words of the question. Some assumptions made in the question are incorrect and should be modified.
The term orbital was introduced by Mulliken in 1932 to denote the one-electron orbital wavefunctions. It cannot be applied to the orbits of the Bohr-Sommerfeld model of the old quantum theory. Bohr's orbits are exactly the same as classical orbits, with the only difference being that they must satisfy the additional quantization conditions on the adiabatic invariants.
Therefore, the orbital is a concept of the modern QM only, without a role in the old QM. In the so-called one-electron approximation of the atomic wavefunction, each electron is in a spin-orbital (i.e., a one-electron wavefunction including the spin part) and has well-definite energy and angular momentum (without any approximation). The number of spin-orbitals with the same energy or angular momentum defines the degeneracy of the corresponding values.
Then, the experimental study of orbitals has nothing to do with counting the number of electrons in a given orbital (that number is $0$ for unoccupied orbitals, $1$ for occupied spin-orbitals. Instead, it corresponds to extracting experimental information about electronic wavefunctions' shape and spatial extension.
Such information can't be obtained directly because wavefunctions are not directly observable. However, the study of electronic density can shed light on this topic. One of the most direct techniques is diffraction experiments.
As an example, there are some experiments where some information about atomic or molecular orbitals has been obtained using these techniques. See, for instance, this paper. Notice, however, that what is really measured is the density corresponding to an orbital.
