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Others have commented on the physical process of interference, waves add up but also cancel out. But the server sound 'disapeaearing' is not caused by interference, in fact it doesn't disappear at all. It's a phenomenon called 'masking' caused by your auditory perception. Louder or lower sounds mask quieter or higher pitched sounds.
Edit: and to add a bit to the answers, natural sound sources add up more than they cancel out on average, so it will get louder the more sources you add. But sound pressure level is also inversely proportional to the squared distance, so there's a limit to the max db you can get just adding more similar sources, with a higher limit with higher source density, ie the closer to each other you put them. I leave the formula for this as an exercise for the reader.
Wouldn't a louder room raise the noise floor, too, so that any quieter signal couldn't be extracted from the noisy background?
If we were to put a microphone and recording device in that room, could any amount of audio processing be able to extract the sound of the small server out from the background noise of all the bigger servers? Because if not, then that's not just a auditory processing problem, but a genuine example of destruction of information.
That's an excellent question, and like most excellent questions I think the answer is 'it depends' but I would bet for 'yes'.
First, 'noise' is a part of the signal we don't want or we don't care about. In op's example they talk about 'two servers' one quieter and another louder. The sounds of these sources is the part of the signal we care about so the sound of the louder server isn't noise in any case. As long as the sound of the quieter server rises above the noise floor and the recording device have the dynamic range to record the sum of the sounds you should be able to 'measure' it (as in pointing to differences in the readings when you turn it on and off) and maybe cancel the other server to some extent by processing it.
Now to this point is a yes but 'it depends' because while the sound of the louder server isn't noise it can rise the noise floor depending on the acoustics of the server room. Some server rooms have lots of reverb and echo, specially when they're not very full, but in my experience most don't, many put these plastic curtains to contain the AC and they dampen sound reflections pretty effectively. But with bad enough acoustics, a very loud 'loud server', and a quite quiet 'quiet' one the sound floor could rise over the quieter server sound.
Disclaimer: this is from the top of my head, I'm tired and not a hundred percent sure this is correct. Don't take any important decisions based in my autistic rambling.