The issue is reported by ASD user "String Bass", who writes:
I installed Snow Leopard yesterday. When I came down this morning I discovered that all of my rogers.com email boxes had 4 copies of each email in each in box. The me.com mailbox did not have this issue. I put them all off-line when it appeared that copy number 5 was starting to download. The box to delete files on the server had ben reinstated for one of the rogers accounts, which had not been set in Leopard. I have used this computer since late May using Leopard without this problem.
The problem seems to persist whether users delete their messages or not, though there does not seem to be a pattern as to when or why the messages download again. The issue seems to be directly related (in most cases, at least) to POP3 accounts and is only present (with the exception of one user on the thread) in Snow Leopard.
ASD user "Ernie Stamper" suggests this workaround for POP3 users:
Those having this problem, please use the Finder to open Home/Library/Mail and locate the file named MessageUidsAlreadyDownloaded3. Use Get Info to see when this file was last updated -- should be after the last time new messages were downloaded?
Quitting Mail and removing this file, and then relaunching Mail will result in a freshening of the file, if it has become corrupted, but will once more download everything still on the server. But hopefully for the last time.
Some users also found success when excluding the same "MessageUidsAlreadyDownloaded3" file from their Time Machine backups. To do this, navigate to System Preferences > Time Machine and click Options. Click the (+) button in the new window to add a file to exclude from backups. Navigate to username > Library > Mail, select the MessageUidsAlreadyDownloaded3 file, and click "Exclude".
There is still no official fix from Apple, though as we continue to mention, Mac OS X 10.6.2 is fast approaching release and may address this issue. Stay tuned to MacFixIt for a complete breakdown of the update when it is released.