No, once you have accepted the insurance company’s offer, you are foreclosed from going back for more compensation at a later date--the case is effectively closed. In many cases, the insurance company will quickly put a settlement offer on the table that is much lower than the true cost of your injuries may be simply to see if you plan to be aggressive about recovering fair compensation. The unfortunate reality is that many accident victims accept these initial settlement offers, allowing the insurance companies to keep their profit margins high.