Lot of head scratching ensued about the fault. Nothing really made sense.
Totally by chance we found out what was wrong. One Saturday when the office was empty and with no a/c background noise one of the other admins noticed the PCs in the problem row weren't making any fan noise. A quick investigation revealed that the PSU fans had been blocked with paperclips and bits of paper causing the CPU to overheat.
An older member of staff who was in that row found the PSU fan noise irritating and just decided to take action to stop it. She genuinely didn't realise she was breaking the PCs.