Your insurer only needs to pay when and to the extent the policy says they do: the policy is a contract, and they are bound by its terms, but do not need to go beyond their terms. If you feel they unnecessarily dragged their feet, you could in theory sue them for breach of contract and breach of the covenant (requirement) of good faith dealings in contracts to get reimbursement for the amount of days they unreasonably delayed, if you can show that the delay was unreasonable. For example, if they should have come by 6 days earlier, you could possibly sue for days' storage charge (though it may not be economically worthioe to do it).Otherwise, the at-fault person would be responsible for the costs, though if another person (not you) were at fault, you'd have to pay the costs up front, then seek reimbursement.