Prior to about the 21st of Feb Italy only reported three cases of coronavirus in the entire country with the most recent case being a Wuhan evacuee at the start of the month: https://uk.reuters.com/article/uk-china-health-italy/first-i... As you say they started to lock down very soon after that. February 24th is very much at the start of the detected outbreak, and it seems safe to conclude that the virus was indeed already circulating widely at that point. They went from basically no reported cases to the most cases in the world outside China and their healthcare system being overwhelmed in days.

While other experts had pointed out that Italy likely had more cases than already reported, the numbers they threw around were more in the hundreds range rather than substantial percentages of the entire population: https://www.nytimes.com/2020/03/21/world/europe/italy-corona... This does explain a lot though, both about what happened in Italy and why containment efforts in other Western countries failed whilst ones closer to China geographically seem to have done much better.