If your diamond engagement ring has started to snag with your hair or clothes, then it means that there is something wrong with the prongs of the ring. In most cases, sapphire diamond rings snag because their prongs have become raised or worn out.
If one of the prongs in your diamond engagement ring seems like a little bit raised above the center stone or if it is not flat, then it is due to a simple prong issue. Maybe the jewelers who mounted the stone in the ring were not able to push these prongs well enough or you might have accidentally bumped the ring somewhere. The good news, however, is that you will be easily able to fix the issue, although you will need to seek the help of an expert jeweler for that.