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: 
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.
**People forget economics**

Why would you want a 5% tolerance resistor? Because it's cheaper than a 1% tolerance resistor and your circuit can stand the variation.

A good example of such a circuit is a voltage divider where *both* resistors are 5% tolerance. Generally (not perfectly, but generally), resistors of the same tolerance will vary in value in the same way under identical environmental conditions. Thus, it's not worth paying for 1% tolerance when all you care about is the unitless ratio of the two resistances.

In this case, if the rebuilt board already had a 5% tolerance resistor, another 5% tolerance resistor would be required or the ratio would vary with environmental conditions, which can be bad.

*Note: This isn't perfect, obviously, but I've not only seen this done, I've done it myself, especially in microelectronic design where you know for a fact that the manufacturing process of the two resistances is identical. Heck, we'd use short strips of polysilicon rather than actual resistors that would take up more real estate so long as they had the same orientation to guarantee the same processing encroachment.*

I can't help it... I'm reminded of an old joke from my college days. A mathematician, a physicist, and an engineer are asked to answer the question, "what is 2+2?" The mathematician proudly answers "4!" The physicist smirks and says "4.00000!" The engineer looks a bit bewildered by their answers and responds, "any damn thing that works."