The military knew in 1998 that Shinseki's estimation of 400,000 soldiers was correct.

Bryan at Why Now? goes back in time:
"This shows where General Shinseki got the number he used for the invasion. He didn’t just pull it out of thin air, it was a number calculated for this war game. It also means that General Tommy Franks knew damn well that the Rumsfeld number was way too low.

"They knew what the problems were going to be, and they did it anyway. These were the problems anticipated with a force of 400,000 troops and they tried it with about a third of that number."

James Fallows in an interview in February of 2004, when they could already see the war was going badly:
"The Army, by contrast, was saying that beating Saddam Hussein was only part of the job. You needed then to think about what would happen afterwards, and towards that end, you needed more people than you would in the smallest possible expeditionary force.

"Therefore, there was a kind of bidding game that went on between the civilian leadership and the Army, where the Army and its allies in the other forces were saying, "We'd like about 400,000 troops to go in." Rumsfeld's idea was more like 75,000. Through a process of negotiation, the U.S. finally went to war with the low 200,000s of troops in Iraq.

"So the bottom line was--?

"The bottom line of the tensions between Rumsfeld and the military was that the force went in at a much smaller level than the uniformed military had been recommending, and a larger level than Rumsfeld would have ideally preferred.

"What was the basic reasoning behind Rumsfeld's philosophy?

"Number one, this had worked in Afghanistan. A very different kind of battlefield, but [that] kind of innovative special forces-intensive approach had won with low U.S. [troop] levels in Afghanistan. Second, it was part of his overall philosophy for a streamlined Pentagon. Third, it was a sense that the Army was just too much like the Gen. George McClellan army in the Civil War -- too cautious, too ponderous, too unwilling to take risks.

"Specific to prewar, what was the main reason the Army wanted so many troops -- why they thought the numbers that they were talking about were necessary?

"The Army had both a specific and a larger metaphysical reason for wanting to have a lot of troops going in. The specific reason was their very precise argument that it would be harder to occupy Iraq than to conquer it. You would need a relatively small number of troops actually to beat Saddam Hussein's military, but then occupying this quite large and quite fractured country would be quite hard, and would take a lot of troops.

"The metaphysical reason was the Army's sense that, once the sort of glory of the short war was over, the people that would actually be there would be the U.S. Army, not some international force, not the Air Force. The U.S. Army would be taking the heat for whatever went wrong. We wouldn't have enough of them there.

"Explain the reasoning of the head of the U.S. Army, Gen. Shinseki, on how important the first days after the fall of Baghdad would be.

"Shinseki of the Army drew not only on his experience in the Balkans, trying to administer a fractious region postwar. [He also drew from] all the corpus of evidence that had been produced by the Army War College, by every other group that looked into this, to say that there was a crucial moment just after the fall of a regime when the potential for disorder was enormous. So there would be ripple effects for years to come, depending on what happened in those first days or weeks when the regime went [down] …."

Update: NTodd has more.