Question: I work at an electronics store and the other day a customer came in who was rebuilding a circuit board. I sold him some resistors, but later he came back in wanting to return some of them because they were only 1% tolerance, and he needed 5%. I'm not an engineer but I have a hobbyist's understanding of electronics, and I was under the impression that there are situations that would call for a more exact value, but never a *less* exact one. Am I mistaken?

Answer: 
If he truly did want 5% instead of 1% then he may have been trying to bin the resistors. That is, individually testing each one and selecting for a particular value.

Are you sure he did not say he needed 0.5%? Because no one says it is "only 1%" then proceeds to get 5%.

Otherwise you might prefer 5% if the resistor composition you needed simply did not come in 1%, but that's preferring one composition over another, not 5% over 1% and I would not expect anyone to phrase it as such.
For a random signal noise source, you may want noisier resistors that tend to come with larger tolerances because of different technology. Admittedly that's extra-handwavy because the additional noisiness of, say, carbon resistors comes from shot noise while a good random signal noise source would rather tend to rely on Johnson noise (which is more dependable).

Another handwavy reason might be that more precise resistors tend to be trimmed, and bein suitable for trimming implies an underlying structure that is more susceptible to stray inductivity.

Note that either of those reasons is really grasping at straws as an intellectual game.

The straightforward answer is "no", really.