You can claim on more than 1 insurance if you have it. Our bags were delayed 5 days at LHR once.

we had amex and natwest cards both with insurance. BA gave us a £250? cashcard each as we went back through JFK and we then claimed from each insurance using different receipts (so 3 different sets of receipts for different items- 1 for BA, i for amex and 1 for natwest)

Some months later amex contacted us as they had tried to claim some back from BA but couldnt as we already had. We showed them that it was for different items and it was fine.