EDIT 2018:
I now have a new Lenovo laptop (also with only USB3 ports), and the problem is gone. So whether it was Intel chipset specific, or just resolved, I couldn't say.
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Here's a bit more of a detailed writeup:
Windows 10 Professional (latest updates)
Calibre 2.77 (problem has existed with all older versions too)
Kobo Aura One (latest updates)
Dell Latitude E7450
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Accessing the Kobo via USB without Calibre running works perfectly. I can browse the file structure, copy on new files, delete files etc. I can leave it connected indefinitely without any issues.
Using the Kobo Windows app also works without issues.
When I run Calibre with Kobo connected (and without any other apps open), after 30 seconds or less I get a blue screen of death with this error:
FAT_FILE_SYSTEM (fastfat.sys)
This happens whether I have Calibre running first and then connect Kobo, or if I open Calibre after the Kobo is connected.
This laptop only has USB 3.0 ports so I cannot test it on USB 2. It happens with all three ports, and with any USB cable that I've tried.
Again, the connection works perfectly without any issues using Windows Explorer. It's only after Calibre opens that the blue screen happens.
I also have a Kindle Paperwhite Gen 3 which does NOT exhibit the problem on the same laptop. It seems to be Kobo/Calibre specific.
I have reset the Kobo to factory settings twice, but it didn't help.
To work around the problem I am closing Calibre before I want to copy books across, and dragging-and-dropping manually in Windows Explorer.