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.
Of course agreeing with all other answers that generally say "no", I see those four cases:

1. In analog musical circuits not-tight values may produce "the tone", when all loose tolerances in filters and other elements sum up, making one's tone less reproducable. Thus, even when you build a clone of your sound processor and give it to a friend, you two sound slightly differently. Though this does not apply to Hi-Fi equipment - rather artistic ones (e.g. guitar effects, synthesizers etc.).
2. It was the point to measure them and prepare some report/statistics - for example for students classes.
3. Preparation for bigger order: as 5%s are cheaper, he searched for best under that price label.
4. Thinking about next serviceman: seeing 1% would think "why it is so tight and do I really need the same for replacement?".
About half a year ago I had to replace 1200 uF capacitor in somehow regular switching power supply. As a hobbyist I was wondering for a while if I really need this value (I had only 1000 uFs in my box and live in a place without electronic store by the corner). I had used 1000 uF I had and since then owner of the equipment hasn't complained - and we are still friends! ;D.

The customer may have not mentioned that being sure that you will provide him with 5%s as defaults.