I'm going to go against the grain and admit that I did buy the extended warranty on a mid-priced dishwasher recently. It was the first time that I've purchased an extended warranty.
My previous dishwasher only lasted me a few years.
The reality about most mid-level appliances nowadays is that they're not built to last... they're only built to last beyond the warranty period.
And replacement parts are not always available, even a short time after the appliance was produced. So the possibility of having to prematurely replace an appliance is pretty good.
Keep in mind that you would pay an appliance repair guy $70-100 just to come and give you his opinion on whether it's worth fixing.
So sometimes I'd rather pay a similar amount upfront in the form of an extended warranty and guarantee myself a longer warranty period.
I can't say whether it has paid off yet... ask me in a few more years.
I don't think that I would buy it all the time, though. It probably depends on the brand, the type of appliance, and what warranty the manufacturer has on key components.
The trick is remembering that you did buy an extended warranty, keeping all paperwork, and knowing how to contact the store in the event of a breakdown.