A molecule absorbs light having a specific wavelength, why doesn't it absorb shorter wavelengths?
Classical approach
Absorption occurs when there are &quot;oscillators&quot; that can resonate with the frequency of the electromagnetic wave. In classical weave physics, these frequencies are called natural frequencies.
These oscillators can involve electrons (UV, VIS, XR), bonds (MIR, FIR), nuclei (gamma rays). They are able to absorb electromagnetic waves changing their spins(NMR, ESR) orientation (microwave), configuration (infrared) nuclear configuration  ($\gamma$ rays) and finally electronic distribution (VIS, UV, X-Ray).
Of course, not all the materials have oscillators that absorb in all the spectrum. For wavelength between 10 nm and 100 pm the oscillators, that mainly involve electrons in these regions, have to absorb energy between $10^{7}$ and $10^{9}$ J/mol without being ionized, in this case only the inner electrons of some atoms (almost always metals) absorbs these wavelengths see XAS, so in our organic world seems that most compounds are X-Ray transparent but in fact there are a lot of inorganic compounds that absorb X-Ray.
When radiation is absorbed there are many ways to release his energy(see figure 1). We can divide them into two great categories radiative relaxation and non-radiative relaxation. In fact radiative relaxation is most common when we are dealing with shorter wavelength because with more energy is possible to excite an electron to a higher level and this can sometimes re-emit radiation when it goes from a higher level to the ground level, this as you know, is called fluorescence and is very common when you deal with UV, and X-Ray (see X-Ray Fluorescence spectroscopy). Eventually, there is also non-radiative relaxation process that are the main processes that occur from VIS (note that however is possible a VIS-NIR luminescence for some compounds) to the other spectral region with lower wavelength.

Quantum approach
Compound energy levels are quantized, so compounds can absorb only some discrete energy values. These values are determinate by transitions that respect strictly selection rules. If you know the energy of the transition, from Plank relation:
$$E=\frac{ch}{\lambda}$$
you can see that for every energy value there is a specific wavelength, so this means that compounds can absorb only at specific discrete wavelength. However if you see a UV-VIS spectrum you will see a continuous spectrum because roto-vibrational changes affect the energy levels of the compounds involved.