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).
depending on the circuit you are making, you don't need the precision. For example, if it is a current limiting resistor it will do the work either at 1% or 5%.

Honestly the main reason to prefer 5% over higher precision like 1% is cost, the other is power, it is very hard to guarantee very high precision for higher power components.

Edit: now for your specific case, I can't see a case where you would want less in general, if everything else is equal then yeah 1% is within the 5% and should work in all conditons.