From my Chase Sapphire Preferred Baggage Delay coverage:

"Baggage Delay must be reported to the Travel Supplier to be eligible for this coverage.

. . . .

"This benefit is payable on an excess basis over and above any amount due from any other valid or collectible insurance or any other form of reimbursement payable by those responsible for the loss."

(Emphasis added.)

Obviously, coverage terms and conditions can vary from one insurer to another.