Do real materials emit radiation at all wavelengths? An ideal black body with temperature greater than 0K emits some energy/radiation at all wavelengths. 
Does a normal body with temperature greater than 0K also emit radiation at all wavelengths?
 A: This is the functional form of the perfect black body radiation's emittance. Notice the logarithmic scale of the emmitance, which means an extremely rapid fall off at the edges of the plots.

Real materials will  follow this outline. Here are calculations showing the tails for various temperatures,which are long, for large wavelengths, but still too small to be measurable.
An example of "real material" that follows approximately the theoretical curve  is the curve from the sun, which shows deviations from the theory, but no peaks outside the measurable intensities in the spectrum. The peaks and valleys, not good fits, are due to the different concentrations of materials and mechanisms in the heat production of the sun.
 
The yellow part is what leaves the sun.
The best fit to the black body curve comes from the cosmic microwave background, which has had billions of years to be homogenized.

Do real materials emit radiation at all wavelengths?

It is a matter of probabilities, and  how long the observation lasts. The theoretical curve allows very small wavelengths  and very long wavelengths. Due to the rapid exponential fall off at the high energy side, it is very improbable for a desk at room temperature  to emit a gamma ray. Long wavelengths are more probable, since the curve falls  off smoothly at long wavelengths, but again, from sheer numbers, more and more improbable the larger the wavelength.
So it is a conditional yes, with the limits at the long side the dimension of the universe and at the short side the quantum mechanical energies available for creating radiation.
