[susan53]

Any time when what is said is potentially a cause of embarrassment for the listener. Eg a request, where they might be "pushed" into having to refuse. Small requests are less likely to be refused and therefore don't need such "careful" handling :
A : Brian, can you lend me €2 ? I want to get a coffee from the machine and I don't have any coins.
The same is true of the refusal:
B: Sorry, neither do I. Try asking John.
Neither person is "threatened" by either the request or the refusal and both can be made directly without either person losing face.

But "large" requests are tricky, and the request/reply forms have to become correspondingly more tentative:

A : Brian I'm sorry but I have a big favour to ask. The thing is... (= pre-request sequence "warns" the listener that a problematic request is coming, which allows them to mentally prepare)
.... I've got to pay my rent tomorrow and I don't have any money left. (Explanation of situation - again "prepares" the listener for what's coming)
Would you mind awfully lending me €500 till Friday when I get paid? (Request phrased much more tentatively, by asking about the listeners hypothetical displeasure at the idea of the request)
B : I'm really sorry Anne... (extended apology to warn the listener of the coming refusal and "soften" it)
... I would love to help but... (expression of hypothetical willingness followed by "but", again to warn the listener that a refusal is coming and to "soften" it.)
I'm afraid that ... (another apology =more warning and "softening")
I can't. (the refusal itself)
I don't have the money (Explanation to justify the refusal and emphasise that it's not a personal rejection)

Compare that with the following - which would be much more "threatening" to both parties :
A : Brian, can you lend me €500?
B : No.