This worked for me:
I had just a few messages in my Inbox folder but its associated file ocuppied way more than 3 Gb (near the 4 Gb file size limit), so I compacted it to remove obsolete information, and it's now less than 50 Mb.
New messages are now well downloaded.
From MozillaZine Knowledge Base about compacting and file size:
When you delete messages in an email client such as Thunderbird they aren't physically deleted. Even emptying the Trash does not get rid of them. Instead they are marked for deletion and hidden from view. They are not physically removed until you "compact" the folder. This is a tradeoff done to improve performance in large folders.
And the same site about file size limit:
Prior to version 3.1 the maximum folder size is 4GB, or as limited by your file system [...]
Thunderbird 3.1 added a a 64-bit offset to the .msf file allowing imap folders larger than 4GB, if the file system permitted large files [...]
This change did not help non-imap folders, for example those in Local Folders, including any offline storage.
So pop email folders file size is still limited.