Don't worry JWP, your question wasn't that daft. In fact, when I worked for a well know long haul airline in the UK, the management said it was OK for one of the pilots to "cat nap" when our schedules changed for the worst and we started to work longer hours. The theory was that with one pilot asleep and the other "on watch" safety wouldn't be compromised. The reality was that at 3am in the dark while suffering from acute and cumulative fatigue and now not having the stimulus of the other pilot,the watch keeper was also very likely to have fallen asleep. The aircraft would require a pilot input every 20 minutes otherwise alarms would go off. I was woken by this on several occasions! Although many systems failures have alarms, some situations such as a storm cloud have none. (I often wonder if this was a factor in the Air France Atlantic A330 crash).
My reply to the management regardin "catnapping" was that I had been observing my cat and have come to the conclusion that it meant sleeping for 20 hours out of 24 on top of the central heating boiler, waking up occasionally for a stretch, some food and a pooh! I could cope with that.