Disclosure: used to be an actuary at an insurance company. I think the general rule is that your insurance will reliably cover the things that it explicitly says it covers, but that category may be narrower than you think, so you should read the policy and figure out what your policy does and does not cover.
For example, if you go to a doctor who's not in your health insurer's network, you're probably not covered. In some states, you can buy a "full coverage" (liability + comprehensive + collision) policy with a liability limit of $10k, so if you hit someone with your car and they have more than $10k in medical bills they can sue you for the balance. Your provider network and liability limit will both be spelled out explicitly in your policy documents, and neither concept is really that complicated, but if you don't know about them you could be in for a nasty surprise.
Of course, all of that creates a giant pain in the ass for consumers, and that begs the question, why doesn't anyone just make an insurance policy that covers everything the policyholder thinks is covered? And the answer is that nobody would buy it! It would be more expensive (usually much more expensive) than its competitors, and people are generally very price-sensitive in their insurance purchases, and no one would read the fine print to see what they're actually getting for their money.
P.S. The rabbit hole of stupid insurance regulation runs deep. Texas county mutuals are a fun example. Captive reinsurers are another.
It's a legacy category of company that gets favorable regulatory treatment (mostly in the form of more flexible pricing) for auto insurance policies written in the state of Texas. It's been illegal to form a new Texas county mutual since 1955, but the ~20 that have remained in existence since that time are still grandfathered in. Some of them have been acquired by big national insurers (e.g., GEICO fully owns one and has a controlling interest in at least one other), which issue policies on the county mutuals' paper and then cede (transfer) the business back to the parent company. Other county mutuals effectively license their favorable regulatory treatment to other, unaffiliated insurance companies. A significant minority of the auto insurance policies issued in Texas today are still issued on county mutuals' paper.
For example, if you go to a doctor who's not in your health insurer's network, you're probably not covered. In some states, you can buy a "full coverage" (liability + comprehensive + collision) policy with a liability limit of $10k, so if you hit someone with your car and they have more than $10k in medical bills they can sue you for the balance. Your provider network and liability limit will both be spelled out explicitly in your policy documents, and neither concept is really that complicated, but if you don't know about them you could be in for a nasty surprise.
Of course, all of that creates a giant pain in the ass for consumers, and that begs the question, why doesn't anyone just make an insurance policy that covers everything the policyholder thinks is covered? And the answer is that nobody would buy it! It would be more expensive (usually much more expensive) than its competitors, and people are generally very price-sensitive in their insurance purchases, and no one would read the fine print to see what they're actually getting for their money.
P.S. The rabbit hole of stupid insurance regulation runs deep. Texas county mutuals are a fun example. Captive reinsurers are another.