"It is that ..." sentences in a non-linguistic context
"It's not that I'm sick. It's just that the water went down the wrong pipe."
The phrase "It's not that..." is the subject of the sentence and is a substitute for a repetition of the phrase or subject that has already been mentioned. If the context is not clear it can easily be misunderstood.
"My coughing does not mean that I'm sick. It just means that the water went down the wrong pipe."