I don't know if this is right, but "The bear has got the potato." is slightly different and might be Bjørnen har fået kartoflen." "I got the potato." is "Jeg fik kartoflen." "Get" can mean "receive" or "acquire". The trick to this is that if you have received or acquired something then you must have it. People often announce that they have something by saying that they have got it, but they are giving more information; that it is something that they didn't always have. Of course, why announce something that you have had for a long time.