It seems pretty unlikely. There are no other reasonable theories that explain or predict such radiation, and it's been studied since it's discovery in great detal. In fact, when Planck gets to it's science we will know even more.
No, see that's what makes the CMBR so unique. It comes from everywhere in the sky (our entire view of the Universe) so can't come from any localized event. And it is incredibly the same in every direction. Almost exactly the same temperature in every direction.
The measurements that have been made to show some structure within it are on the level of millikelvin. If you had your monitor show a pure white screen, the variation in brightness is millions of times more than the variation in that background temperature. There are few ways to explain such a source coming from all directions.
The other unique feature of the CMB is that its spectrum is a pure blackbody ie produced by a hot object.
The origin is not quite the big bang itself, but the universe as a whole when it was a hot (very nearly) plasma, protons and electrons, some He, Lithium nuclei. In this state light of all wavelengths could not travel far before interacting with an electron or nucleus. Because of all this constant interaction, the universe was (very nearly) all the same in all places, and of course the same in temperature. As universe expanded it also cooled (spreading energy over greater volume). When the temperature fell low enough the electrons combined with the nuclei making atoms. Light (all wavelengths) could now travel more-or-less freely across the universe. The CMB is the light from the moments just before this combination time. This temperature is very hot (can't recall number) and what we see is the red-shifted version of that blackbody (thermal) radiation.
There is a very neat observation. The CMB interacts with a neutral carbon and cause its electrons to switch between "hyperfine" levels emitting a radio photon. The neat thing is by measuring the radio spectrum of distant gas clouds it is possible to detect these photons and measure the temperature of the CMB as seen by that cloud, Distant clouds should have higher temperatures in line with their redshift, This is what is observed, so the CMB is consistent with being reshifted light from the "edge" (time of formation of atoms) of the big bang. This kills the "tired light" model.
Example observations in the paper Nature vol 408, pp 931-935 2000, R. Srianand, P Petitjean, and C Ledoux.