Why does signal averaging reduces noise levels by more than $\sqrt{n}$?
Well, I would say the assumption that your noise is Gaussian is ill fitting.  If the noise is due to machine interference, it probably has some tonal characteristics.  Tones of the same frequency can reinforce or cancel each other out when added.

To get a better idea of what might be going on, you should:

1) Make a histogram of the noise

2) Take an FFT of the noise

The histogram should look like a bell curve and the FFT results should be flat.  If they aren't, your noise is not Gaussian.

Hope this helps.

Ced