If you make the cloud thicker and thicker, the small flux between the lines will rise until it too becomes optically thick and gets truncated at the black body curve. After all this, you might be forgiven for wondering if there are any optically thick objects in the universe. And indeed, most gas clouds will be optically thin, except perhaps at the wavelengths of certain key emission lines. There are, however, plenty of things up there in space with such high column densities that they are optically thick at pretty well all wavelengths. Stars are one example.
Planets are another.
The early universe, back when it was still ionised and hence opaque to light, was also optically thick, which is why the microwave background has a blackbody spectrum. I'll leave a full discussion of this to quantum mechanics and statistical thermodynamics, but here is the basic idea. Thus at long wavelengths, the spectrum rises quickly towards higher frequencies. Luckily, something suppresses the very high frequency emission, and this something is quantum mechanics. For really high energy emitters, they are so far apart that the thermal energy cannot get any of them up to even the first excited state.