Context. Dust emission at sub-millimeter to centimeter wavelengths is often simply the Rayleigh-Jeans tail of dust particles at thermal equilibrium and is used as a cold mass tracer in various environments, including nearby galaxies. However, well-sampled spectral energy distributions of the nearby, star-forming Magellanic Clouds have a pronounced (sub-)millimeter excess. Aims: This study attempts to confirm the existence of this millimeter excess above expected dust, free-free and synchrotron emission and to explore different possibilities for its origin. Methods: We model near-infrared to radio spectral energy distributions of the Magellanic Clouds with dust, free-free, and synchrotron emission. A millimeter excess emission is confirmed above these components and its spectral shape and intensity are analyzed in light of different scenarios: very cold dust, cosmic microwave background (CMB) fluctuations, a change of the dust spectral index and spinning dust emission. Results: We show that very cold dust or CMB fluctuations are very unlikely explanations for the observed excess in these two galaxies. The excess in the Large Magellanic Cloud can be satisfactorily explained either by a change of the spectral index related to intrinsic properties of amorphous grains, or by spinning dust emission. In the Small Magellanic Cloud, however, the excess is larger and the dust grain model including TLS/DCD effects cannot reproduce the observed emission in a simple way. A possible solution was achieved with spinning dust emission, but many assumptions on the physical state of the interstellar medium had to be made. Conclusions: Further studies, with higher resolution data from Planck and Herschel are needed to probe the origin of this observed submillimeter-centimeter excess more definitely. Our study shows that the different possible origins will be best distinguished where the excess is the highest, as is the case in the Small Magellanic Cloud.