Q: If someone has left their employment and has cobra and is now going to start a new job that does not offer health insurance can they still stay on their COBRA? A: Yes, you can still stay on your COBRA coverage until your 18 months is over. You can continue COBRA for any reason, except if you get another insurance policy through an employer or an individual policy. Also, once you are eligible for Medicare COBRA coverage is no longer available to a person.