Does a second similar source of white noise increase the overall noise level? If I put next to my cooler another similar cooler that produces similar white noise, will the overall noise level increase?
I want to point out that I am speaking about adding another independent, non-correlated white noise source, not just another pair of speakers that produce sound of the same phase and curve. It is evident for me that adding another speaker will increase amplitude twofold compared to one speaker, but this is not evident for coolers which produce non-correlated sound.
 A: 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

ans =

   10.0410   10.0832

>> mean(([(w1+w1) (w1+w2)]).^2)    % add one of them to itself, add both

ans =

   40.1640   20.6295


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.
A: The white noise of both coolers will add up and appear to be one white noise with stronger intensity than the white noise of each individual cooler.
The reason is that the white noise produced by each cooler can be seen as a sound wave that has a large amount of "random walk" behaviour to it.  The two similar white noise sources will both have a "random walk" behaviour but as they are just similar but otherwise not related to each other, they can both (at some points in time) cancel each other or (at some other points in time) amplify each other in some direction.  When they work against each other and when they "cooperate" is, well, random.  So the result when combining them is essentially... white noise.  However, as they sometimes "cooperate" and result in wave spikes in the same direction, you end up having bigger spikes and alterations back and forth in the combined white noise than in the individual white noises, so you end up with a more intense white noise.
(IIRC, the formal definition of white noise is such that if you add two white noise waves with the same frequency together, you end up with a result which mathematically is white noise)
A: Yes. The noise from the two different sources should be uncorrelated, so they add in power. Adding a second identical noise source increases the power by a factor of 2. Since the power is proportional to the square of the amplitude, that means the amplitude increases by a factor of $\sqrt{2}$.
In terms of decibels, this means that the power increases by 3 dB, and the amplitude increases by 1.5 dB. (Power is more commonly measured in dB than amplitude, but there's no reason you can't use dB for both power and amplitude.)
A: What you very probably have in mind is noise cancelling. If you have two sounds which are exactly the same, but one of them is inversed (phase and anti-phase) and you play them together, they cancel each other out. The problem is that having two exactly same sounds in real life is almost impossible, without using any microphones and/or digital processing that is.
Also, to have two sounds with aforementioned qualities cancelled out, you'd need to listen to them together perfectly, that means they'd need to point to your ears in a way so that you listen to the phase without any offset to any of them. This is, again, in real life, impossible.
So your idea is based on something solid, but it can't work in your case, because (to sum it up):

*

*the two sounds of the coolers are different

*the two sounds of the coolers are not in an inversed phase to each other

*the two sounds of the coolers aren't pointing perfectly to your ears :)

Therefore, your two coolers cannot cancel each other out, nor can they together produce the same level of volume. The volume of course increases.
sengpielaudio.com says:

20 dB gain modification should give the ratio of 4 (four times) for sensed volume and loudness
20 dB gain modification gives the ratio of 10 for measured voltage and sound pressure and
20 dB gain modification gives the ratio of 100 for calculated sound power and acoustic intensity
Doubling of the volume (loudness) should be felt by a level difference of 10 dB
Doubling the sound pressure (voltage) corresponds to a measured level change of 6 dB
Doubling of acoustic power (sound intensity) corresponds to a level change of 3 dB
3 dB = twice the power (Power respectively intensity - mostly calculated)
6 dB = twice the amplitude (Voltage respectively sound pressure - mostly measured)
10 dB = twice the perceived volume or twice as loud (Loudness nearly sensed - psychoacoustics)

