Hello dwhaas,
... and welcome to our forum.
The only time that I have seen such an issue is if the user has two different user profiles in Active Directory. Once they used the one SharePoint was looking for the user was able to open and save the form.
Do you have a "Windows Server 2003" Domain. In this case I have seen that you have to do the following settings for single user.
Internet Explorer:
Tools > Internet options > Security tab > Custom Level
then scroll down to "User authentication" - "Login"
then activate "automatic logon with current username and password"
Hope that helps.
Regards,