Sometimes with difficulty. Sometimes not. One has to remember that caring is a part of loving. You could not care if you do not love (at least to some degree). Love is not a choice. It is, however, a great motivational force.
“When you've cared too much and you get sick and tired of certain people's mean words and actions,”
This is a slightly different issue. The “sick and tired” is a reaction to the “mean words and actions.” It is very difficult to care for someone who is mean to you, especially if they are really mean and trying to hurt you. It might sound so easy to just walk away and never care again. But love does not work that way. Even if we walk away, we still care. Even when it seems that we can not take any more meanness, we sometimes find ourselves unable to just yell STOP.