I don't think its necessarily the wood construction that sets the height. Here in Miami there is no wood construction at all but we still see the 5-8 story apartment buildings that go up everywhere. They are just all concrete.
See this:
https://www.google.com/maps/@25.7352...7i16384!8i8192
or
https://www.google.com/maps/@25.6933...7i16384!8i8192
or
https://www.google.com/maps/@26.1280...7i16384!8i8192
Here are some 7-story ones under construction so you can see the concrete structure:
https://www.google.com/maps/@26.1299...7i13312!8i6656