There's an old saying that "no [man] is an island". We are fortunately or unfortunately connected to those around us and they are affected by what we do. Especially in the case of two parents quarreling and affecting the children, in the case of employers making decisions that affect their customers and employees, and in the case of two neighbors fighting over property rights. Most of the time when people take the time to think about it they realize that their actions have sometimes serious consequences on those around them. They may not even realize it until there is an emergency.
Anytime there is a serious legal conflict stress is high. It is very easy for everyone to get into their respective corners and refuse to budge or to admit that they made a bad decision. However, being willing to admit that to yourself and discuss it with your lawyer or a mediator is a good step towards resolving an escalating conflict and saving time and money.
One of the reasons that mediation works well is that it encourages parties to a conflict to "slow down" and try to resolve issues. Compromise and creative solutions are strongly encouraged, not seen as signs of weakness. AND the procedure is confidential.
Refusing to back down no matter what the financial and emotional cost is a hard way to go. There may be a few situations in which that stance is absolutely necessary, but there are also a lot of times where that stand leaves no winners.