1. Being involved in a serious accident, where the other driver is uninsured.
If you have been hit by an uninsured driver, and you don’t have a UM coverage on your own policy, you are pretty much out luck as far getting compensated goes, except for your med-pay and health insurance policy that will provide for the medical expenses, and property damage coverage for your damaged vehicle, if you had that kind of coverage on your policy. You will not be compensated for your past and future wage loss if you are unable to work due to your injuries, and pain and suffering / emotional distress associated with the injuries suffered.
When you have an uninsured motorist policy, you don’t need to worry about whether the other driver has valid insurance. You will have your own sufficient policy that will cover you for whatever amount you are insured for all your losses.
2. Hit and Run Accidents
Some of the worst accidents are caused by hit-and-run drivers. People who cause serious accidents due to negligence or due to driving drunk or due to both have an incentive to try to escape liability and/or criminal charges by fleeing the scene and hoping that no one will ever find them. Many of these hit and run drivers are indeed never found and prosecuted. Your uninsured motorist coverage will also cover you in the hit and run situations.
The above scenarios should make it clear to you that you must not save on your policy by not having sufficient uninsured motorist coverage, as the risks are way too high, while the potential savings are relatively insignificant.