Firstly, if copying the book across without calibre involved doesn't work, then the problem isn't caused by calibre. The problem is going to either be the device or the connection from the PC.
This could be a database corruption on the Kobo Touch. The fix for that is a reset of the device.
But, as adding books from the Kobo site works, it might be something else. The possibility is that you have a corrupt or malformed epub on the device. Each time the device processes new books, it gets to the corrupt one and stops. If you know what books you added when this started happening, you could remove them and see if the others are processed properly. If that doesn't work, or you can't remember the books, you should remove all the books that the Kobo doesn't have in it's library (delete them while connected via USB), and then add them a few at a time until you find the one that was bad.