From Reader George:

Snip

The number one reason we don't shoot down North Korea's missiles is that we cannot.

Officials like to reassure their publics about our defense to these missiles. Japanese Chief Cabinet Secretary Yoshihide Suga told his nation after last week's test, "We didn't intercept it because no damage to Japanese territory was expected."
That is half true. The missile did not pose a serious threat. It flew over the Japanese island of Hokkaido, landing 3700 km (2300 miles) from its launch point near North Korea's capital of Pyongyang.

The key word here is "over." Like way over. Like 770 kilometers (475 miles) over Japan at the apogee of its flight path. Neither Japan nor the United States could have intercepted the missile. None of the theater ballistic missile defense weapons in existence can reach that high. It is hundreds of kilometers too high for the Aegis interceptors deployed on Navy ships off Japan. Even higher for the THAAD systems in South Korea and Guam. Way too high for the Patriot systems in Japan, which engage largely within the atmosphere.

All of these are basically designed to hit a missile in the post-mid-course or terminal phase, when it is on its way down, coming more or less straight at the defending system. Patriot is meant to protect relatively small areas such as ports or air bases; THAAD defends a larger area; the advanced Aegis system theoretically could defend thousands of square kilometers.
But could we intercept before the missile climbed that high? There is almost no chance of hitting a North Korean missile on its way up unless an Aegis ship was deployed very close to the launch point, perhaps in North Korean waters. Even then, it would have to chase the missile, a race it is unlikely to win. In the only one or two minutes of warning time any system would have, the probability of a successful engagement drops close to zero.
"When over Japan, they are too high to reach," tweeted astronomer Jonathan McDowell, in between tracking the end of the Cassini mission. "You'd have to put the Aegis right off NK coast to have a chance."

"It's actually virtually impossible to shoot down a missile on the way up," adds Gerry Doyle, deputy business editor for Asia at The New York Times. "Midcourse or terminal are the only places you have a shot." That would mean for a test missile shot towards Guam, THAAD would have a chance to engage, though it has only been tested once against a missile of this range. For the test flights over Japan that would mean the only engagements possible are to the east of Japan, when the missile was on its way down. But there is little reason and huge logistical difficulties in having U.S. Aegis destroyers and cruisers loiter in the ocean there, waiting for a possible test launch.