Aha! This was the clue that 'fixed' my similar problem, but with a different module and with DNN 3.2.2.
I had installed a dotnetjedi counter on a child portal and noticed that sometimes the images of the digits were not displayed and gave the red X. That was an aggravation, but not fatal. However, at the same time, users suddenly had to repeatedly login again when trying to navigate the site. That was a MAJOR aggravation.
The problem of being logged off was not limited just to tabs with 10 or more modules, it occurred with any number.
I deleted the dotnetjedi counter and the login problem immediately resolved.
So, maybe the common thread with some of these repeated login issues is related to the images.
Jill