Two personal examples:
Some years ago, I was involved in a highway crash during a cross-country trip a long ways from home. Nobody in either vehicle had any injuries, and while my vehicle was totaled, his large commercial vehicle was barely damaged. He was quite pleasant at the time, joking as we waited for the cops, and I didn't think anything of it. So, imagine the gut wrenching feeling I had when two or three years later, I get a random lawsuit in the mail. As I soon learned, the statute of limitations was going to be up in a matter of weeks before they filed the suit. The suit was pure bullshit. Claimed he couldn't work, etc. Also got sued by his piece of shit wife who claimed loss of company, or something like that--basically a legal maneuver for them to double the lawsuit by saying the wife (who was not in the accident at all) is being hurt by my actions. Utter bullshit. Unfortunately, at the time, I had a $50k limit, and I was now being sued by two parties each seeking damages "in excess of $50,000". I honestly don't know if that excess of $50k is just a legal thing that puts it in a different court or something like that, or if that number was tied to the fact that my insurance limit was $50k. Anyways, I was mortified that my limit had been so low at the time and I was scared I was going to be on the hook for hundreds of thousands of dollars. So what happened: it slowly got drug out (as all lawsuits do I think) over the course of 2+ years. My insurance company provided the lawyer, who I talked to a couple times on the phone, including a taped deposition. I also filled out some papers describing exactly what happened. Thankfully, I never had to travel to the state in which I was being sued (and my lawyer was based).
Eventually, I got a phone call from the lawyer saying the insurance company settled for $30k total. Quite frankly, I wanted to counter-sue for the pain and suffering I legitimately had during those two years of fear and unknown. It was just a lump in my stomach, something I couldn't shake, keeping me awake at night. But of course, what I wanted even more was to never be part of a lawsuit ever again.
Moral of the story: Your car insurance company should have your back if you're sued, and your limits will probably cover it if they get a settlement, but then again, maybe not. If I had caused actual serious medical issues, he would have been in his right to sue me for a lot more and it wouldn't look so frivolous anymore, and he could have possibly (and rightfully) gotten way more than my insurance would pay out. Since that time, I've always kept my car insurance at a $250k/$500k limit, and I carry a $1 million umbrella. If I got sued for something frivolous again, would they get more than $30k out my insurance company since I have such high limits? Maybe, but I don't really care as long as it is under my limit. The savings of cheap, low insurance went right out the fucking window the moment I was sued by the high cost of living with fear of losing everything. While I'm sure I would have still had some restless nights had my insurance limits been maxed, I doubt it would have been the same in this particular example.
A few winters back, I slipped on some ice on a sidewalk in front of a person's house. Goofed up my knee in the process of falling. MRI, lots of PT, and lots of medical bills that my health insurance company had to pay. Sometime during this process, I get a letter from my health insurance company asking for details about where exactly I fell, whose house I was in front of!, and how it happened. To me, it seemed they wanted to know if they could somehow recover damages from the person whose house I was in front of who should have kept the ice clear from the sidewalk. The form never said anything specifically about that, but that is how I interpreted it. Now, technically the homeowner was at fault because I did slip on ice in front of their sidewalk, but my God!, it was my own dumb fault for falling. I was definitely not thinking of a lawsuit at all (although in a different, uninsured world, I can see how maybe I would think it....MRIs aren't cheap, people), so in the form I said I couldn't quite remember where exactly it was, which wasn't untrue. It was right around the property line between two houses, and when I fell and recovered I wasn't concerned about where exactly I was in relation to an unmarked property line. Thankfully, I never received another form or other followup about it. Had I put down an address though, who knows what would have happened...probably nothing, but maybe not. This experience did not change how I handle my insurance, but again, I already had the umbrella policy in place.
We (in the US) live in a litigious society. I try to do my part on both fronts by covering my ass with insurance and also taking personal responsibility for my own actions.