Found a solution on my own at
http://help.wugnet.com/office2/insert-OLE-documents-Access-2007-ftopict835047.html
I have found the source of this bug. After going through the Windows
Debugger and logging all Access API calls, I determined that for some
reason, ole32.dll searches for a registry key
called HKEY_LOCAL_MACHINE
\SOFTWARE\Microsoft\Internet Explorer\ActiveX Compatibility
\{00000000-0000-0000-0000-000000000000}. If this key does not exist,
then things continue normally and eventually the file will be
attached. If this key does exist, however, Access will give the error
message. Deleting that key solves the problem, and attachments can be
made.
Upon further research, is appears that Spybot: Search and Destroy
(among other programs) puts this key in there. I am curious as to why
Office 2007 uses this key when previous versions of Office did not. In
any case, if anybody from Microsoft is monitor these forums, this may
be something to add to your Knowledgebase, as I'm not the only person
who has had this problem. I have the logs of the API calls if
necessary.