While it is wise to obtain new car insurance before canceling an old policy but once does obtain the new policy one needs to cancel the former policy with the old insurance company. Many states require a policyholder to send a written request to their insurance company to cancel their insurance policy.
If a person does not pay on their policy and has not requested their policy to be canceled then in most states the insurance company must send a notice for nonpayment. In your son's case he can try to now cancel with his former insurance company and provide them of proof of his current insurance coverage with his new company to see if they will go back to the date of the new coverage to cancel.
If possible your son does not want his old insurance provider to report that his old policy was canceled for non-payment because that would be reported and placed on his credit history and make it difficult in the future to obtain insurance.
So while in many cases an insurance company might stop your policy for non-payment it is not good for you or your credit history if this is done. For more information on your state's auto insurance laws and the state's guidelines for insurance companies check with the insurance regulator for your state.