Since I got it on April 24th, my Apple Watch did not connect to my WiFi network at home.
Apple claims that the watch works without Bluetooth connection to the iPhone as long as they are on the same WiFi network.
Reality
For me, the WiFi connection at first did not work. I checked my DHCP server but it claimed that it has never seen the Apple Watchs MAC address.
I tried to restart the iPhone and the Apple Watch and vice versa but the watch refused to work without Bluetooth connection to the iPhone. It did not even show up in any network traffic.
Solution
My WiFi requires MAC addresses to be registered and the first thing I verified was, that there was no typo in the address of the watch and the registration was correct. The MAC can easily be found in the information of the watch itself.
Finally I tried to remove the WiFi network from the iPhone (Settings > Wi-Fi, tap the blue arrow of the WiFi in question and choose “Forget this Network”). After reconfiguring the WiFi again and turning Bluetooth on the iPhone off, the Apple Watch instantly popped up on the network and kept working as expected!
2.4GHz vs. 5GHz WiFi
In case you have a 2.4GHz and a 5GHz WiFi at home, Sébastien Page wrote a very helpful article named How to make Apple Watch play nice with your iPhone on a 5Ghz Wi-Fi network over at iDownloadBlog.