Good question. Assuming the cosmic background radiation is coming from the temperature of hydrogen gas within the last billion years, we would expect to see it gathered around visible galaxies. This is not the case, as the cosmic background radiation is nearly uniform in every direction. Furthermore, the intensity of the cosmic background radiation at 2.725 kelvin happens to fit perfectly with the big bang theory's prediction!
Another way of answering you is to say that the cosmic background radiation IS being attributed to BOTH "the big bang" and to "the emission from excited hydrogen". You see, about half a million years after the big bang, the universe had cooled enough for it to be transparent to light. When this happened, photons generated by hot hydrogen ions (read: protons) were created in abundance. These photons are still out there to this day, on their long journey to end up in one of our telescopes.