The issue seems to be related to the "Read More" link that can be added to posts.
Do you use any plugins that may affect the default behaviour of the Read More link, or does your theme include functions that may change the way Read More is displayed?
One way to check all of this is to deactivate all other plugins, one at a time, and switch to one of the default themes for a few minutes.
Could you also make sure that you only use one instance of Read More in your post?
Let me know if you find a conflicting plugin!