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Although any competently designed microphone preamp should function perfectly when used within its capabilities, sound engineers have a habit of not always being sufficiently careful. Here's a scenario where things can go wrong...
Suppose a microphone is being used to record natural-sounding speech. With the speaker at a reasonable distance from the microphone, the output could be around -60 dB most of the time, with respect to the kind of signal that the mixer or recorder needs, which is around one volt. So the engineer applies 60 dB of gain.
However, although speech has a mostly consistent average level, it is also very 'peaky'. There are frequent transients that vastly exceed the average level, often by as much as 30 decibels. So when these come along, as they will, they are also amplified by 60 dB. If this kind of level were amplified correctly it would reach over thirty volts! But in practice, there will simply not be enough voltage available in the preamp's power supply range. So the signal is clipped.
This clipping takes place over a very short time period. However, at the moment it occurs, the preamplifier's ability to correct distortion, through the technique known as negative feedback, disappears. So there is a momentary and intense burst of distortion. Also, the preamplifier is now destabilized and will take a certain time to recover. All of this is audible, and since different designs will react in different ways, they will certainly sound different - some better than others.
Ideally, the preamp would have metering that would detect this and make clipping known to the engineer. However, most mic preamps have only rudimentary metering, some have none at all, and others have metering that measures the level after the damage has been done! There is clearly some improvement necessary here.