In some states there may be a grace period permitted by law for a person to pay their car insurance but in many states it is left up to the insurance company if there is a grace period or not. Where it is left up to the insurance carrier many do not have a grace period but instead insurance cancels at 12:01 am of the day that the policy expires.
State laws differ, as do insurance companies' guidelines, regarding if there is a grace period between a due date and an insurance policy cancellation. In some states there may be a grace period or a number of days in which the company must give you notice of cancellation. For example, in New Jersey if your insurance company cancels your policy for non-payment you should receive a 15 day warning notice.
Most states require a notice of cancellation so many insurance providers include the notice as part of the payment notice or mailed bill, stating that if the payment is not paid before a certain date the policy will be cancelled.
If the cancellation/bill states 11/1/06 it will likely list a time element saying 12:01 AM meaning that at midnight, the day before, the policy will be cancelled if the payment is not made. Many companies have a toll-free number on the bill to call if you have billing question. Do not wait until the day of cancellation to call and get information or it may be too late.
To find out if your state has an insurance law regulating a grace period or a time period in between when your auto insurance is due and when you pay that allows you to not be canceled, check with your state insurance regulator. If the state does not have a law then check with insurance company to find out what rules they have regarding this and their cancellation process.