The error message is usually correct, but it's not impossible that you are running into a problem with a newer version of Acrobat. Try the following: Open one of these documents in Acrobat 8, and without doing anything else, close the document again. Does Acrobat prompt you to save the file? It should only if the document was modified. Sometimes Acrobat is able to repair such a corrupt document without any user interaction besides a popup that is displayed for just a spit second that says something like "Repairing PDF document". The only evidence you have of that is that you need to save the file. If that's the case, can you then open the newly saved file in Acrobat XI?
If that's not the case, it's still not clear if the file is corrupt or not. Only an analysis of the file would give you that information. Have you tried to open the file in Acrobat or Reader DC? If that works as well, then I would say that there is something wrong with Acrobat XI.
There are no prompts when closing the document. Also the documents do not open in DC. There are multiple older files that are no affected directly by this issue also. Hosted on the web server.
What are the steps to fix any issues with Acrobat XI? As we believe this to be the issue.
The chances are you have a corruption or error in the PDF that Acrobat XI is sensitive to. is there any file you are able to share publicly? The smaller and simpler the better.