There are two types of coverage on your auto policy, liability, and physical damage (that would be collision and comprehensive). In most states, the liability will carry over onto a TEMPORARY SUBSTITUTE vehicle, such as a rental car. Physical damage does NOT necessarily carry over, though.
In either case, LOSS OF USE for the rental car, while it's in the shop, is NOT covered under your policy. In my experience, if you return a car with a scratch, the "loss of use" time for repair is about 3-4 days, and your credit card WILL be charged for that 3-4 days, EVEN IF THE REPAIR IS NEVER MADE.
So, primary insurance FOR THE NAMED INSURED will carry over in most cases, for liability coverage. In many cases, also for collision and comprehensive (although some companies like Progressive specifically do NOT transfer physical damage to a temporary substitute vehicle). But loss of rents is NEVER covered.
You'll have to check with your credit card company, but I've never heard of one covering loss of rents, either.
Now, the bottom line - I do NOT recommend my clients buy that waiver coverage they like to sell, UNLESS you're renting a vehicle out of the country. Then, the whole international thing is just too much of a pain to deal with (I'm NOT going to tell you horror stories, but it's bad), and just to save the $5 a minute international charges to Europe, you're going to want to buy it.
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