Hi Fred,
IMAP accessed email accounts are often downloaded partially. the mails we see in the Apple Mail are often a blend of the mails from the downloaded mail copy and partly from the server. Accessing without internet is the exact status of the emails. Time Machine Backup only backups emails from the local caches of the Apple Mail. thus, you can expect incomplete emails in the backup.
for the same, we can't blame just Apple MAil, as the same is seen with Outlook and Thunderbird. while their is no permanent solution for this so far, Mail Backup X helps it resolve it to a great extent.
The application marks those incomplete emails as "Uncached Data Warning" and has the option to resolve it on its own. on some cases, it guides the user to download the emails forcibly.
later, accessing the exact email from the corresponding mail programs, you can find the emails complete with their attachments and contents.
I would recommend the same. Still, take a free trial before going for the full version:http://www.mailbackupx.com/download.html