Well, no telling how long they'll take to respond to my trouble ticket, but in the meantime I did some sleuthing and uncovered the problem: there's a bug in how the Obi's wi-fi handles WPA2 keys.
My key starts with a non-alphanumeric character, and Obi cannot handle that. Interestingly enough, the nine other devices on the same network, from multiple manufacturers, have had no problem at all with the same key. I stumbled across the bug when I tried my guest network, which WPA2 key starts with an alphanumeric, and got in.
I altered the first character of my primary key to be alphanumeric, and the Obi was able to connect to the primary network.
In the case Obi doesn't fix this bug, maybe this will help prevent someone else from all the headache it has caused me. If and when they ever respond to my trouble ticket I'll tell them about it.