Try a new/different SD card in the camera and take some test shots and try to import them. If they import ok then the SD card you were using has failed and corrupted the images on it.
Not thinking its a card problem as Lightroom 5 reads it just fine, just not in Lightroom CC but thank you for the suggestion.
What type of files. Maybe your install of LR 6 is corrupted. Uninstall and Reinstall. You do that from the CC desktop app.
What operating system are you using?
Using Mac OSX Yosemite, Reinstalling may be my next move.
Assuming that Adobe did not fix the error message from LR5, the actual problem may be WRITE issues.
In the import dialog, check your output and secondary directories for typos in the folder names.
Also check to see that you are read and write privs on the destination folders.