Yes, you can register your name as a trademark provided the Trademark Office of your country or the country for which you are seeking protection considers it "distinctive." As to whether a name is distinctive or not depends on a variety of factors. As a rule of thumb, the more common the name is the less likely it will be considered distinctive as there may be many others with the same name. Likewise, the more unusual the name is the greater the distinctiveness and the greater the likelihood that registration will be granted.
In any event, even if you have been refused registration of your name as a mark either for the reason that it was not considered distinctive or because someone else had already registered it you are not prevented from using your name in the course of your business for ordinary business purposes.