I am not sure, just give me a moment
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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So, basically, if I the contract reads that they are not responsible for any damages that my occur over the course of my rental term, and I have signed saying that I accept these provisions then I am SOL? Are there any other routes that I can take with this? I will have to review the contract to see what it contains but if that is the case, is there anything else that I can do to follow this up?
Yes, if the rental provisions preclude them for any sort of liability, and the insurance doesn't cover it, then you unfortunately seem to be out of luck. You could always try suing but your chances would likely be pretty low since these storage places (including parking garages, etc.) often draft the contracts to protect themselves from any sort of liability, even if they're clearly at fault.
If such is the case, you could try an internal appeal through the insurance to plead for them to reconsider, but if that fails, then again it comes down to litigation and if there's something in the contract precluding them from coverage of this sort then your chances at recovery are slim.
But in any event, yes it would be wise to thoroughly review the contract to see what recourse may lie within it.
I know that's probably not what you were hoping to hear but hopefully that is of some assistance.