I assume you're talking about "coolers" as in CPU-coolers or other cooling systems in computers. The noise-spectrum from such a fan is not white strictly speaking, you can see it measured on this page. The spectrum is fairly level if you look at small parts though and ignore the tonal components.
Anyway, by your definition, two uncorrelated white noise sources add without interference. The "peaks and troughs" (sound pressures) don't add up coherently (in ideal sources of course :), but they don't cancel each other perfectly either. Therefore the sound power is doubled. By definition the RMS-amplitude (proportional to the sound pressure) is then multiplied by $\sqrt 2$. The sound intensity is proportional to the sound power and hence is doubled as well.
I actually simulated this in Matlab, because the terminology with sound power, sound energy, sound levels, sound pressures etc is daunting, misleading and confusing.. I'm specifically avoiding talking about decibels.
>> w1 = wgn(8192, 1, 10, 'real'); % generates real white noise, 8192 samples
>> w2 = wgn(8192, 1, 10, 'real');
>> mean(([w1 w2]).^2) % show mean powers of both noise signals
>> mean(([(w1+w1) (w1+w2)]).^2) % add one of them to itself, add both
So yes you do get a doubling of power when adding uncorrelated white-noise sources.. and when perfectly adding a single source, you quadruple the power. By definition the RMS amplitude is the root of the above.