Today I discovered the same "Lightroom has encountered problems reading this video..." issue described above. GoPro MP4s and also AVIs that can be viewed and played in the Finder, but the LR front-end cannot read.
This issue sort of happened out of the blue. I have been importing at ton of files from Aperture. Sort of happened out of the blue. LR was pretty rock solid before. Maybe I should reimport these video clips? Whaddayareckon?
In December I downloaded and installed this LR5 update: Lightroom_5_LS11_mac_5_7-2.
Today I deleted Lightroom then reinstalled Lightroom_5_LS11_mac_5_7-2.
I also trashed the "lrdat" and ".plist" files and later purged the video cache.
In my case, the issue remains unresolved:
Ironically, I am in the process of moving from Aperture to Lightroom because APERTURE could not read AVCHD files captured at 60FPS on a Sony RX100. Funny. A non-standard frame rate apparently and no serious software updates form the Mother Ship.
EDIT: Bloody LR5 is STILL purging a 3GB video cache. Slow. The USB2 cable linking the drive to the computer is a painful bottleneck.