The short answer is it depends. Of course, your insurance company may choose not to renew your policy for many reasons. Among those reasons are tickets, accidents or other information which demonstrates that you are a high-risk individual who is likely to make a claim. (Remember insurance companies only make money by taking your premiums without ever having to pay a claim on your behalf).
Generally, if you are not at fault in an automobile accident, and your insurance company pays you monies under its policy, they will not usually drop or increase your coverage.
If however, you are at fault in an accident, then your insurance company may attempt to raise your rates when your coverages are renewed. In rare instances, and after several claims or tickets, your own insurance company may drop your coverage or increase your premiums to the point that you will have to seek new coverage.
Certain types of coverages, like uninsured motorists coverage and medical payments coverage, provide payment to you even if the other driver is at fault, but does not have sufficient coverage to pay all of your damages. Making a claim against your own insurer for these types of damages under these types of coverages do not provide a basis for your insurer to drop you.
Your insurance policy also requires that if you are involved in an accident you must cooperate with your own insurance company and assist them to a reasonable extent. Failure to provide this cooperation can result in a denial of insurance coverage thereby leaving you personally exposed for damages. Thus, even if you are at fault in an automobile accident, it is prudent to report the accident as soon as practicable to your own insurer.
When a person makes several claims within a certain period of time (usually three years), the insurance company may not choose to drop you, but instead put you into a "pool" of other high risk drivers for whom the premiums are substantially increased. Some insurers will double, triple or even quadruple the the premiums for such high risk persons. Alternatively, the insurer may only offer insurance in minimal coverages to avoid the potential of paying a large loss on behalf of someone they feel is a bad risk.
The best way to avoid these consequences is to be a defensive driver. Always be aware of the other guy. Most insurers reward drivers who do not make claims with premium discounts.