From what I've read, the most reliable period to shoot down these things is on their down-trajectory. At which point it was probably already no longer over Japan.

edit: for instance this Reddit comment https://www.reddit.com/r/japan/comments/6wmnxp/north_korea_f...

> The THAAD system that is deployed in Japan intercepts an incoming missile as it is coming down on the target, with a maximum altitude of about 150 km.

Which gets you to reading https://en.wikipedia.org/wiki/Terminal_High_Altitude_Area_De...

> Terminal High Altitude Area Defense (THAAD), formerly Theater High Altitude Area Defense, is an American anti-ballistic missile defense system designed to shoot down short, medium, and intermediate range ballistic missiles in their terminal phase (descent or reentry) by intercepting with a hit-to-kill approach.