Are they both 64-bit? Just a wild guess…

Both are 65 bit

64 bit

What is your account type on both machines - POP or IMAP?

Both are IMAP

Do you have 'Synchronization' enabled for the account on both computers? Synchronization means downloading an offline copy of each message to your local disk.

I have some folders checked for synchronization on my desktop, but not all. I'll check the folders on the laptop at home later and see if they are similar. However, the problem is only with the Inbox folder, not the whole account. The inbox folder is different in size on the 2 computers.

Question: should I have synchronization on or off?

That's up to you. I'd always have it turned on, because it essentially allows you to backup your account.

Will reply again tomorrow after I check the laptop settings tonight.

I'm at the laptop now, and changed the synchronization settings to the same as those on my desktop. I checked synchronize all messages regardless of age on the laptop, and TB proceeded to download a lot of messages for awhile, and while it did the size of the Inbox declined in real time, and now it matches the desktop Inbox size of 61 MB exactly. So, this fixed the problem, but why? If all messages are now on the laptop too why did the Inbox size shrink? This makes no sense to me. But it worked!

May be it was a combination of downloading and compacting. http://kb.mozillazine.org/Compacting_folders

When your problem has been fixed, can you mark the thread as 'Solved' please? Thank you.

The problem is fixed, but I don't see a button to mark the thread as solved.