While a lot of the answers focus on cost to replace and how much money you should have for tangible goods. There are a few more issues to consider.
However before we get started, these issues are not related to ones net worth. They are related to other factors. Having money certainly helps, but someone worth only $10 may not need to insure their stuff under some circumstances.
Insurance is a risk avoidance strategy. As such, it should be used to avoid risks that would otherwise cause issues for you. The normal example is a house. If you lost your house due to fire, would you be able to "make it" while you paid the mortgage off, and got a new mortgage to pay for a new house? This is a relatively simple view, but a good one.
These days people tend to look at insurance as a savings account. I payed in X so I am entitled to Y. Heath insurance (a bit more on this later) is exacerbating the issue by selling it's self that way, but it simply isn't true. What your paying the premium for to avoid the risk of loss. Not so you can have a pool of money to draw from in time of need, but so that a time of need should never arise.
Which brings us back to, should you get insurance?
Let's assume you have no legal or contractual obligation to have insurance. If you put the money you were spending aside would you have enough money to secure a new asset should your current one just vanish? This is the normal argument. But it has a second side. Do you need the asset at all, or can you just accept the loss. Lets pick on a red neck for a second. While certainly not millionaires, or "well off" by conventional means, the guy with 6 cars on bricks in his lawn does not need to insure 6 cars. If one were to vanish, it may make a hardship but hey, he's got 5 more.
So with tangible goods it's more of a question of can you afford to replace the item, do you need to replace the item, and how big a risk is it to you to loose the item? What would you rather loose, the item, or the cost of the insurance?
I am going to try to keep this as un-rant like as I can manage, but be aware that I am biased.
There are two big examples of non-tangible assets that are commonly insured. Life Insurance, and Health insurance. There are others, but it's very hard to get people to pay money to insure something that they don't actually have. Ideas can be insured, for example, but in order to insure an idea you have to spell it out, at that point why not just file for the patent etc. etc.
Keep in mind that a lot of people and companies will insure against losses due to IP theft or other such intangible things. Largely these follow the same rules as tangible assets. This section is meant to focus on those insurances that do not.
Life insurance is a bit odd. Were all going to die, so it seems like a "good bet" but what your insuring against with life insurance is an early death. For term life insurance it's a gamble. Will you die before your term runs out. For full life insurance (with no term) it's a different gamble. Will you die before you have paid in what they agreed to pay out. In many cases it's also a gamble that you will miss a payment or two and cancel the policy before you die. If the risk of your death worth the insurance. Usually while young the answer is yes. Do you leave your Family short one earner? Will they make it without the insurance? But as you get older, as life insurance becomes more of a sure thing it also becomes less needed. Your kids move out, there not dependent on you any more. You have retirement accounts setup so your partner need not worry should something happen. What risk exactly are your trying to avoid at this point. You will die. You have planned for that eventuality, it's not a risk anymore, it's a fact.
Is another beast all together. Historically you insured against some catastrophic event, that you couldn't really plan for. Say a heart attack. Surgery and treatment would run in the tens of thousands, so it would ruin you if you didn't have insurance to cover that. That was the risk that you were avoiding. A big, expensive event, causing financial ruin. However, over time it has shifted into something else. The general concept is still there, insure to avoid a risk. But the "risk" has been widened to include all manor of things that are not actually risks.
For example a flu. You would go to your doctor, pay your co-pay, and your insurance would pay the rest of the visit. Then you would go to the drug store and get the drugs, pay your co-pay and the insurance pays the rest. But what risk, in this instance are you insuring against? That you can't cover the cost of a doctors visit? That you can't cover the cost of the medication?
In this example, a common one, historically the "mother of the house" would go you have a flu, have some chicken noodle soup and go to bed. That would be the end of it. Cost of care is a day's lost wages (or maybe a weeks) and a few cans of soup. However today, because we choose to, the cost of care is much higher. We go to the doctor, pay our co-pays, the insurance has to pay it's part. The doctors office has to carry the cost of the staff it takes to see you, and the staff it takes to handle the claims with the insurance company. And now your flu, cost $1,500. But again that's not exactly true either.
With heath insurance and "normal" medical care (like sprained ankles, and colds, etc.) the insurance only really covers the cost of having insurance. In that same flu example, if you went to the doctor as a "self pay" (no insurance) you would often time get a much lower, and reasonable rate. Frequently, under the cost of your standard co-pay. This seems like the doctors being "bad" but it's not. They don't have to file a claim, they don't have to keep track of it. They get immediate payment, not payment 6 months down the line that they need to share with other businesses.
With "critical" or "catastrophic" care, heath insurance is still a good thing. If you have a big, unforeseen event, then heath insurance is great at helping you avoid that risk.
With chronic (long term) care, your back in the same boat as the flu. Often times you can get better, and cheaper, care as a self pay patent, then as a insured patent. That is not always the case however. So you have to measure your own circumstance, and decide if insurance is right for you. But remember insurance is about risk avoidance, and not about paying less. You will ALWAYS pay more for insurance. It's designed that way. Even if the cost is hidden in many ways. (Taxes, spread out over visits, or prescriptions, etc.)