their policy. Then, more than likely, they'll craft a story to demand that full amount. The higher your limit, the more they will want of it because insurance money is easy money.
Your state likely has minimums and, in many states, you can't be sued under many circumstances if you meet those requirements. However, in terms of your personal liability beyond your limits, remember that your liability depends on what attorneys are willing to pursue more than the damage you caused. You can always be sued beyond your policy limits even if you don't have any money or if you have all the money. Practically, though, attorneys rarely pursue lengthy, costly lawsuits against properly insured people, even those with a assets (assuming they are not very wealthy/public figures). Though they can sue anyone.
Carrying higher insurance limits is wise—accidents involving multiple injuries can quickly surpass even high coverages. Pay a bit for higher limits like 250/500 so you can sleep at night, then hope and pray your kid doesn't find some black ice into the back of a school bus. If you have a lot of money then an umbrella could make sense. But if you have 10M in assets and buy a 10M umbrella then cause a 20M medical bills accident, you're back where you started.