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Jeff OJoe - your question is a good one, and the default answer is that how you are compensated for an item will depend on the language in your insurance policy as well as the laws of your state.
There are two ways to "value" personal property in an insurance policy - ACV (actual cash value) and RCV (replacement cost value). ACV is the depreciated value of the item. For example, you may have a 5 year old TV set for which you paid $1,000. Under an ACV policy you will receive the value of a 5 year old used TV - maybe 50% of what it would cost to replace it with a comparable model today (which may cost less than your original purchase price). If your policy has RCV coverage, one of two things happens. Either you get the value of a new, comparable item, or you get the depreciated value up front and can recover some or all of the withheld depreciation if you replace your item, depending on what you pay for it.
One thing to keep in mind - the replacement cost of your item is not necessarily what you paid for it, but what it would cost to replace it with a comparable item today. That HD TV you paid $3,000 a few years ago can be replaced by a like kind and quality model today for $1,500, so that will be the replacement cost of the item. The same holds true for jewelry - if you paid $5,000 for a ring and the insurace company can replace it for you with a ring of the exact style and quality for $3,000, then they only owe $3,000.
What about antiques and collectibles? In general, they are automatically valued at ACV. This may seem contrary to logic, but the reason is that they don't depreciate, so in effect you should get the "actual cash value" of the item immediately. A five year old TV might only be worth 50% of a new TV, but your T206 Cobb is worth exactly what it is worth on today's market, even if that is more than you originally paid.
As an added wrinkle, an insured is not supposed to profit from his loss. This could raise a question with collectibles. Suppose you have a card graded PSA 5 that you paid $1,000 for a few years ago, but that same card today is worth $1,500. An argument could be made to support you only receiving $1,000 up front for the loss of your card, with you eligible to receive an additional $500 if you actually go out and replace it for $1,500. After all, you shouldn't "profit" from your loss by not replacing your item and still receiving more than you originally paid for it. This also raises all kinds of tax questions, as the added value of the item probably constitutes a capital gain.
Collectibles are very tricky business in insurance claims because, unlike most types of personal property, they don't lose value over time. I can tell you that as a former property claims adjuster, I hated dealing with claims involving lots of antiques and/or collectibles. Determining value is often difficult, though eBay has made it much easier - in the event of a loss involving collectibles, it's probably one of the first places your adjuster will look.
Jeff