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I think the American Airlines and United Airlines cards are the only two airline cards that offer the coverage. If you google "Trip Delay Reimbursement and the Credit Cards that Offer It" you will find a nice article on this topic by The Points Guy.
According to the most recent benefits guides I could find (which may not be up-to-date):
Citi AA Executive delay coverage kicks in after 3 hours.
Citi AA Platinum takes 12 hours.
United Club takes 12 hours.
No card is the best at everything when it comes to travel insurance. Any decent coverage involves getting an AF card. Some Citi benefits (like delay protection) on Prestige kick in more quickly than corresponding Chase ones do on CSR, increasing the frequency they'll prove useful. Other Citi benefits (like cancellation and interruption protection) aren't very useful when using miles.
Personally, I use Prestige.
@Anonymous wrote:
Thanks for the replies. Hmm, so it's probably best to use a regular card that offers the protection rather than get an airline card without that protection I'm thinking.
Currently have Chase Ink Preffered and CSP which both offer overnight delay/cancellation insurance.
Keep in mind that with United, the fine print says (and I believe it's enforced) that you have to use the card to pay for a ticket to get the free checked bag benefit. Most cards don't require that.
Also, if you're close to reaching elite status, then elite-qualifying miles from heavy use of an airline card may be useful.