I think the perfect example of this is with parents and children. Many times children, particularly teens, often see things very differently then their parents. This can easily turn in to the misconception that one or the other of them is wrong, which can lead to arguments and fighting. The fact of the matter is that for each of them, this is their reality.
Let's take a situation where a child or teen is very busy, doing sports or involved in after-school activities. The child may eventually become exhausted and overwhelmed and start refusing to go to meetings or practices, or acting out at home. From the child's perspective, they are saying that they need a break, and they are becoming burnt out. From the parents viewpoint, they are saying, that the child has made a commitment to a team or a club and that they need to live up to that commitment. On top of that there may be a financial obligation that the parent has made. Is either one of them wrong? No. Both the parent and the child are right, they just have different realities. The problem continues when the child, unable to see the parents reality, feels that the parent is pushing them too hard, and doesn't feel heard or understood. The parent, unable to see the child's reality, may feel that the child is being uncooperative or lazy, and feels that they are not being respected or listened to by the child. For both parties, this is how they are feeling. I tell clients all the time, your feelings are your own. No one can dispute them or take them from you.
This is why in conflict resolution, it is so important discuss things in terms of feelings as opposed to accusations. In an argument, when emotions are heightened, it is very easy to say, "But you're just being lazy!" or "Why won't you listen to me?" Instead, saying something to the effect of "I FEEL like you are having a difficult time with your schedule." or "I DON'T FEEL like you are understanding what I'm trying to say." might be more productive. When you explain how you feel, it becomes very difficult to dispute. This will often also lead to more meaningful discussions, and better understanding on both sides.
