Borderline rant approaching in the form of a less polished version of Leslie's advice:
My explanation would be: "Any insurance the bank buys is for the benefit of the bank, not for the benefit of the renter. Whether your property is in your living room or a safe deposit box you rent from us, we do not insure it." Having equally plain language in a contract, as Leslie advises, is far better than relying on even my terse verbal explanation.
If I realized that a safe deposit attendant was trying to be more serice oriented and was attempting to explain the A vs. B coverage, we would be discussing his career plans. Customers who get anything more than a plain and simple, "We do not insure your property" will not remember all the qualifiers and caveats. They will inevitably say they were only told it was insured, not about the conditions precedent to the coverage.
The safe deposit function is not appreciably more dangerous than several other bank operations and I am sensitive about training techniques that promote an inordinate amount of fear in bank personnel. However, this is an area where careless verbal comments can become warranties and the bank can become the de facto insuror.
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In this world you must be oh so smart or oh so pleasant. Well, for years I was smart. I recommend pleasant.