It doesn't necessarily matter if the representatives are 'adamant' that they're not liable. What does matter is what is said in the storage contract.
If the contract states something along the lines of "[Storage Co.] is not liable to any extent for any loss or damage to property stored, howsoever caused", then you are likely out of luck if insurance does not cover it, since the storage company has waived any liability and you agreed to such upon signing the storage contract. However, if there is no such clause in the contract which abated the storage company's liability, then you should be able to claim against them the amount for which you're out of pocket. You could strongly argue that, in agreeing to hold your properly, they owe you a duty of care to take care of your property and ensure it does not get damages. By letting mice in, they have breached their duty to take proper care of your goods, and this breach caused you damages (monetary and property loss). In other words, you could argue that they were negligent. Alternatively, you could argue that they breached the contract to properly store your goods, since they allowed your goods to be damaged.
In any event, if there is no waiver of liability on their end, you could claim against them the monies you are out of pocket or the damage you suffered, as well as interest and legal costs.
Hopefully that is of assistance.
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