That's a bit simplified. More details about this story can be found in the very entertaining book "Masters of Doom" (about John Carmack and John Romero).
This is an excellent book. It's not just about Carmack and Romero (although it focuses on them) but also about id Software as a company, from its inception through to a few months before Quake III came out, if I remember correctly. A very insightful read.
Loved this book. Bought a few copies for friends to hopefully inspire them about what a small, scrappy team of talented individuals can achieve (and the fun... and pitfalls along the way).