The two Civics we purchased last month were the first two cars that I can remember (other than my wife's leased Toyota) for which we did not purchase an extended warranty on. When we were first starting out, we bought used cars that were still under warranty, then bought the extended to take the vehicle to 100,000 miles. In a couple of cases, I'm glad we had them. Oddly enough, it was on the vehicles that we bought new where we really needed the warranty. This time, we opted to rely on Honda's reputation rather than take an extended warranty. That said, we are putting money into an account to help cover any unforeseen problems down the road. I'm just hoping this decision doesn't swim around to bite us.
Why did we buy the warranties on the other vehicles? Like others have said . . . peace of mind. Plus, we figured that rather than having to come up with the money to fix an unexpected repair post-warranty, we were essentially paying for an insurance policy financed with the car at a lower interest rate, as compared to having to cover the cost of the repair on a credit card, with a much higher interest rate. The increase in the monthly payment really wasn't that much.
My father used to work for a GM dealership as their head mechanic. He told me then, even when rust was more of an issue for a vehicle, never to purchase the rustproofing or "paint protection" options.
When we went into the business office to purchase our Hondas, these things were offered, but NOT pressed upon us at all. More of a mention of options which were available, but that was it.