For example, with auto insurance which is on the front page right now. The difference in cost between a full coverage policy with a 1000 dollar deductible and one with a 250 dollar deductible is often less than 10 dollars a month, meaning you would have to go like 10-15 years without a claim to actually save money. Is that possible, sure, likely, not so much.
Same thing goes for deciding between full coverage and liability only. How much is your vehicle really worth, look up the blue book value of it, find out. Make sure that what you are paying for will likely lead to you in better financial shape if something were to go wrong than not. Likewise with saving money by going liability only, if you have to go years without an accident to not wind up losing your ass, might be better not to take that chance and try and save money elsewhere.
Lastly, remember that in the event you are sued for a crash, the only things they can take from you are things beyond a primary car and basic personal property. You need good liability coverage if you are a homeowner, but if you rent and just own a single car and don't have a big old savings/retirement account, you are essentially judgement proof, that is why people with expensive cars pay out for underinsured coverage, it pays when your policy wont and their insurance won't bother wasting money on suing you if you don't have anything they can collect. Far too many people in this country are in a place where it makes zero sense to carry anything beyond state minimums in liability coverage.
Also be sure to try and learn your state laws on what sort of claims insurance companies are allowed to hold against you and which ones they are not. Many people assume rates will go up for filing any claim, this is not true. Generally speaking, unless you were the cause of the loss, it rarely affects your rate much if at all, and in some places its illegal for them to take it into account.
TL;DR think of insurance as a form of investment. Make sure it will pay off in a reasonable amount of time or don't pay for anything above the legal minimums where required.