No, there aren't any holes like that in the EM spectrum.
There are other ways of creating photons than by having electrons bound in atoms transition from one level to another. (For example, you can create pretty much any frequency of photon you want by accelerating a free electron.)
The quantization of energy levels appears both in quantum and classical mechanics, and it is not a consequence of the Schrödinger equation.
It is a consequence of confinement.
In fact, anytime that a wave equation (any quantum equation for the wavefunction, or a classical equation for a classical field, e.g., EM field) has periodic boundary conditions in some spatial variables, the system exhibits quantized energy levels.
As noticed in the question and in other answers, energy levels in quantum systems are not always quantized.
On the other hand, also classical systems exhibits quantization of the energy levels. For example, consider the allowed frequencies of a string with fixed length (confinement), as in a guitar or violin. In such a string, the allowed "energy states" corresponds to frequencies (harmonics) which are multiples of a fundamental frequency (first harmonic).
In the quantum realm, energy levels are quantized if the wavefunction is confined in a finite space, e.g., in an atomic orbital or in a quantum well. In a solid, energy levels are also quantized, but the difference $\Delta$ between levels decreases as the system size increases. Therefore in the thermodynamical limit (large system sizes), these quantized energy levels become a continuum of states, since $\Delta\rightarrow0$.
As an example, let us consider a plane wave
$$\psi(r)\propto e^{\imath k r},$$
which describes the wavefunction of a free particle (or the propagation of a sinusoidal wave of a classical field). The wavefunction has a continuous of energy levels $\omega\propto k^2$. However, if one confines the wavefunction in the segment $[0,L]$ one has that $\psi(0)=\psi(L)$ which gives $e^{\imath k L}=1$, and therefore the only wavenumber $k$ allowed are $k=2\pi n/L$. Hence, the energy levels of the confined particle are
$$ \omega\propto \frac{n^2}{L^2}.$$
The gap $\Delta_n$ between energy levels goes to zero for $L\rightarrow\infty$. Therefore, if the particle is confined ($L<\infty$) the energy spectrum is quantized (finite and discrete energy levels, $\Delta_n$ is finite). If the particle is not confined ($L\rightarrow \infty$) the spectrum is continuous ($\Delta_n\rightarrow 0$). In real solids, $L$ is typically huge with respect to the typical sizes of the ion lattice, and therefore one is in the limit $L\rightarrow\infty$.
Best Answer
In liquids and solids the difference in energy between energy levels becomes very small, due to the electron clouds of several atoms bein in very close proximity of one another. These similar energy levels will form 'bands' of indistinguishable spectral lines.
In gases however, atoms will be spaced loosely enough such that the interaction between atoms will be minimal. This allows the energy levels to have sufficient difference in energy for distinct lines to be formed.