An insurance policy is written to cover definite start and end periods, and if the policy is terminated earlier than the contract date, it will only be after a written notice is sent to you to indicate the final date of your coverage. At that time, you will either need to have another policy in place or be prepared to pay any costs out of your own resources.
From the legal perspective, insurance companies are not required to give you any grace period at all. If an insurance company chooses to do so, that is a company choice and the length of the grace period is up to their discretion. As far as the law is concerned, it is your responsibility to keep your insurance policies active.
Most insurance companies will provide you with a 30 day notice of intent to cancel. The notice may contain information on how you can bring the policy back into force, such as paying any premiums that are in arrears, but the company is within their rights to cancel your policy if they do not choose to continue insuring you.
Some types of policy are more accepting than others. For example, you may be given a longer period of time to bring a life insurance policy up to date, or even given options that allow the policy to pay its own premiums during a personal financial hardship. Car insurance, because it is higher immediate risk coverage, does not usually provide you with any recursive options and once the policy cancel, the coverage terminates immediately.
The best option is usually to start shopping as soon as you get the notice of intent to cancel. If there is no obvious way to reconcile the differences with your insurance company, try getting a free online insurance quote from a brokerage website like this one, where you will receive multiple quotes, each from a different insurer, so that you are able to make the most informed buying decision.