This is an interesting problem which also appears in gamma spectroscopy. Both 1D NMR and normal gamma spectroscopy are related. A sample emits photons, the photons are then captured by a detector.
In the case of a modern FT NMR machine the nuclei are excited by a radiofrequency pulse, the nuclei in the sample then re-emit radiowaves and the sample relaxs back to the way it was before the pulse in the free induction decay. After capture of the signal a analouge-digitial converter (ADC) supplies it to a computer which then uses the fourier transform to convert it into a histrogram of intensity against frequency.
Gamma counting uses something similar a energy dispersive detector records a series of events, these events are then fed into a ADC whose output is recorded by the computer. Normally again as a histrogram of events at different energies.
In both NMR and gamma spectroscopy there are two sources of noise, I define noise as unwanted signals in a system or spectra.
If we use gamma spectrscopy as an example for a moment, we have random noise due to cosmic rays (things like muons) and effects like Compton scattering. This will cause the random appearance of events in the detector with random energies. This will have a similar effect on the appearance of the spectrum to the random noise which exists in the electronics of a NMR machine. If we were to lower the noise level of the amplifers in the signal path before the ADC in a FTNMR machine we would be able to record spectra faster.
The more times we run the experiment the smaller the effect of the random noise will be on the final spectrum. I once ran an experiment for three weeks to get a spectrum from a lump of trinitite I bought off ebay. It turned out to be genuine.
The question is about the measurement of a stable sample. When I was a PhD student I worked on some samples which were unstable. I suspect to this day that I managed to dissolve for a short time somethings in my NMR samples which then later formed a superinsoluble solid. Once I saw a peak appear in the FT-NMR spectrum at the expected chemical shift which then disappeared again. We can consider this side question also.
Now one problem I see in both systems is if some non random unwanted signal enters the spectrometer before the ADC.
Consider for a moment if we were to do NMR spectroscopy in a 1:1 mixture of CHCl3 and CDCl3. We would have a very strong chloroform peak, this would limit the sensitivity of the amplifiers which can be used before the ADC, by forcing the system to reduce the senstivity of the amplifiers before the ADC what will happen is that the effect of random noise in the ADC will become greater. This would make it harder to make the measurement.
Also with a NMR spectrometer we have a limit as to the number of bits in the computer memory which handles the output from the ADC. If we had eight bit memory. then when one of the points in the free induction decay was to reach 11111111 then the system should not attempt to pulse the sample to record any transient.
If we were to pulse the sample again and then try to increase the count in this point (in time) in the FID, then we would have a nonsensical situation which would cause mayhem with the fourier transform. I think it could then fail to work as intended
Also in gamma spectroscopy we would have the problem if we were to overflow the memory storing the output of the ADC we would start to get a distortion of the spectrum. I suspect that in gamma spectra the effect would be less. We would merely get the highest peak to become zero again. But I suspect that the rest of the spectrum would still be OK.