If you have next business day support, Dell’s support strategy is to delay you as much as possible, asking you to perform unrelated diagnostics checks, until it’s past 17.00. From that point on, they accept the possibility of a hardware failure, and let you know that Next Business Day only applies if the ticket has been confirmed before 17.00 (which you need the diagnostics output for). After 17.00, it’s the day-after-tomorrow.
Apparently this is standard. I should have known.