To be fair, this isnt just restricted to pet insurance
Say if you paid direct debit for your car insurance, you made your deposit payment and then day 2 of your annual policy your car was stolen - essentially you have "used" the policy but you have only paid 1 month of your 12 month contract - generally in the case of car insurance they will deduct the cost of your annual policy from your settlement.
I think as its a pet its more "emotional" - I have to applaud Tesco from Caroline's post, well done them.
I dealt with pet policies a while ago and would always write a personal note when sending out cheques