I think the first two concepts are not that strange: "I am having a cup of coffee" is true and only true. "I am having a cup of tea" is false and only false.
But also "neither false nor true" is very common. For example if you have sentences with a wrong presupposition: "My cup of tea is cold". Since I don't have a cup of tea, it makes no sense to say if it is cold or not. It isn't true that it is cold. But also it isn't true that it is hot, or warm or has any given temperature at all. Also there is some debate if sentences about the future can be right or wrong. What about "Tomorrow the sun will shine"? Sure: Once it is tomorrow, you will know if the sun is shining. But then that sentence isn't saying anything about the future any more, but about the present: "Now the sun is shining." That wasn't the question. The question was about "Tomorrow the sun will shine". Is it now true or false?
Also "true and false" sentences are not that rare: If you are on a train, it is perfectly true, that the station is moving away from you. Someone at the station would object: He is not moving away, you are moving away. In special relativity there are dozens of examples like these. All of them not abstract at all but very commonplace.