This must be a problem with the firmware itself, not with CM. CM is merely a front-end for modifying the collections database, every other functionality is provided by the Kindle FW. Consequently, there is no CM-specific log file.
It is an unfortunate fact that the Kindle software is pretty fragile and is sometimes behaving in very strange ways (e.g., put a book on the Kindle that it doesn't like, and suddenly, all books disappear), and there doesn't seem to be an easy fix. Most of the times, the "Windows approach" (reset everything and start from scratch) seems to work, at least for some time - until this or another problem appears again.
Sorry for not being able to provide a more specific or more satisfactory answer. It's not my software that is misbehaving, it's the firmware - and nobody really knows why.
PS: It may still be a good idea to update CM to the latest version. And possibly - at your discretion - the FW as well. But note that 5.3.6 is a horrible piece of crap IMO, so maybe stick with 5.3.4.
Last edited by ixtab; 07-08-2013 at 05:18 PM.