Only time I've ever had a problem of this sort is thhis past New Years.
I went to Rome and my jacket was stolen in Testachio.
Had my credit cards in it but my passport was in my inner jacket so I was OK there.
When I got to the airport to check in for my BA flight to London they asked for the credit card. I didn't have it so they didn't issue me my boarding pass but instead sent me to the Cust Serv desk.
Gentleman at the desk asked what happened and then asked me for the police report. I produced the PR, he called over to the ticket counter, I walked back over and everything was fine.
I don't know what would have happened without that PR. They were pretty adamant about having the card I booked with.