The story I read or heard was from an interview with Knight many years ago. While I recall that there was a clear reason in his mind that he threw the chair, I am probably not 100% on all the surrounding facts from the story. How true the entire story is is up for personal interpretation, but it was from a writer or journalist (I know, I know) who talked to Knight about it.
He was trying to get in front of the NCAA (I think, or maybe just the B1G), and the powers that be would not grant him a meeting. From Knight's POV, it was because they knew he was right about whatever he was pissed off about, more than likely an instance or 12 of cheating from another school. So, he said, he knew that if he was to be suspended for some infraction, he would be granted a meeting. When he yet again disagreed with an official's call(s), he threw the chair to make sure he got tossed from the game and subsequently suspended. When he got the meeting about the suspension, he took the opportunity to state his case on whatever the other issue was. In his POV, he was forced to do something drastic because the chiefs would not grant him a meeting on the subject he wanted. (That piece of the story sounds very, very plausible, imo.)
I'm no fan of Knight's because of his bullying and off-the-wall temper brought on by his own perceived demons, way beyond what I believe to be reasonable. He was/is a brilliant coach, very smart, cared about his players in his own way, did many good things for people behind the scenes, trusted very few people for his own reasons.
I read the book about their 1987 Championship season by John Feinstein way back when. I remember thinking that it elevated my opinion of Knight, showed the good side of him that few people knew about, gave insight to his coaching during practices and games, and his ongoing internal struggle with anything less than perfection. Not long after, Knight came out and blasted the book as being very anti-Bobby Knight and distorted and made him look bad.... the exact opposite take away I had as an objective reader.
Another favorite story about him is the one where a recruit from Chicago who signed with Illinois after heavy recruiting from IU was killed prior to his freshman year. Knight visited the kid's mother to express his sympathies. (The Illinois coach had not, if I recall.) Anyway, the Mother says to Knight, 'What should I do about the money?' What money? 'The money they gave me.' Who gave you? 'Illinois. Wait here.' She went into a bedroom and came back with two shopping bags full of cash. 'They gave me this money for son to sign with them. Should I give it back?' No, said Knight. Keep it. It's yours.
I have heard same . The kid was Ben Wilson.