Hmmm.. It might be something to do with Code Mirror and CB together. I was using Code Mirror as the editor. When I switched to "No Editor" it saved it.
Just so you know I saved the field, noticed the description had been blanked... so I refilled it... tried both Apply and Save (several times)... and it came back blank each time. Oh... and it's not just the Multi-Checkbox. It happened on several other fields as well... All of them saved when I switched to "No Editor".
Sorry... I should have added that I have been using Code Mirror only for my articles, and haven't had a problem with it at all, except for saving the custom CB fields. I didn't try "Tiny" so have no idea whether it saves them or not.
Ok... but I do consider Code Mirror to be the default Joomla 1.6 editor . It's one of the two editor options that ships with Joomla and "Tiny MCE" is much too buggy (in J 1.6) for my taste .