SBBusboy said

The restored backup works fine on most accounts, but there are two which have this problem. I think the messages are there, but they will not stay displayed to be clicked on and opened. Once the headers have been displayed and have disappeared I have to restart Thunderbird in order for them to display again. I can send and receive messages, and newly received messages remain displayed.

Try repairing the two "Inboxes" on those two accounts.

First: Right click on the Inbox folder

Select Properties

On the little Panel that pops up: Click the Repair button.

Do it once for each of the "Broken" accounts. Let me know if that helps.

(See these screenshots:)

No, that didn't fix it. The Message headers don't even display now. I fear that I've lost my messages...

Try "Rebuilding the Global Database" as per this very useful page: https://support.mozilla.org/en-US/kb/rebuilding-global-database

I hope your backup of your profile folder is not corrupted ... that wouldn't be too good ...

https://support.mozilla.org/en-US/kb/profiles-where-thunderbird-stores-user-data#thunderbird:win8:tb68

One other thing to check ... log into the account through Web Mail and see if all of the messages are in the "Trash" folder still, in there.

You might be able to still restore the messages from the Trash Folder IF they haven't been in there too long ... Select All, and then Drag and Drop them back on the Inbox folder.