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: 
There might be a hidden specification in wanting 5% - the circuit might be relying on carbon film (typically 5%, "beige bones") vs metal film (typically 1%, "blue bones") resistors. Carbon film has different behaviour under temporary overloads (in a bad design, a metal film resistor might fail) and looks (which might be relevant if someone is restoring vintage equipment and wants their resistors as beige as they were).
Yes. If you are building a run of (say) 100 RC oscillators and you want them to have a spread of frequencies from (say) 95-105kHz. So you buy a lot of 5% resistors and **select** to get that range.

Of course there *are* better ways to do this, but it isn't an impossible scenario.

In other words, the customer is always right.

(Just playing devil's avocado here.)