tl;dr: Fire TV devices can only use channels 36–48 and 149–161 in the 5GHz spectrum; they cannot use the Dynamic Frequency Selection (DFS) of the 5GHz spectrum, which spans across channels 52–144.
I've been tweaking my network at home and noticed something odd after reassigning the channels used on my wireless access points: my Fire TV devices were no longer connecting over the 5GHz spectrum, and instead were only connecting over the 2.4GHz channels. I had scanned the environment to find the least congested channels in my area, and found an unpopulated 80MHz–wide one on channel 122.
After I reprovisioned my access points, they were now operating with much less interference from the neighbors. However, bandwidth to my Fire TV clients was abysmal: only 60–80Mbps. When I checked my network settings, I saw the Fire TV devices were only connecting to the 2.4GHz network, and would not connect to the higher bandwidth 5GHz spectrum.
I did a little searching, and found that Amazon has made it so their devices will not connect over the middle–range DFS channels (52–144) in the 5GHz band, only connecting to channels on either ends (36–48 and 149–161) of the spectrum.
I'm posting this in case anyone else seems to be having WiFi issues with their Fire TV devices, and can't figure out why performance is so horrible, when there ought to be ample wireless bandwidth available. Also, the higher end channels (149–161) seem to be the least populated at the moment.