Bohr’s Model could explain it only for Hydrogen atom because he knew that the proton and electron were fixed and could explain the energy of the electron easily.
Another important factor is that the interaction is only between the proton and the electron. If another electron comes in between, it's hard to detect the position of the electron, and the distance between them, so you cannot predict the energy of the electrons. The Bohr model uses quantization of the energy of the electron, but was not based on a theory for that quantization. The quantization of angular momentum followed from the Bohr's allowed orbits hence allowed energy levels for the electron.