The time at which your insurance policy expires can differ depending upon state laws and insurance guidelines. In many cases a car insurance policy will expire at 12:01 AM on the expiration date.
This is often confusing to most people because you expect to have coverage on the expiration date, but in fact the policy expires on the first minute of that day. Therefore, you need to get alternate coverage or renew your policy prior to the expiration date.
So if you fail to pay the premium by the due date your policy is usually cancelled. Some states and/or insurance companies may allow for a grace period but most do not.
Your policy and/or renewal paperwork should state the exact time. For example I just received my own renewal statement and it clearly states that if the policy is not paid then it will expire at 12:01 a.m. on the due date.