Mail has stopped downloading messages from server

[ronincats]

I run OS 10.5.7 on an intel iMac. A year ago April, my hard drive crashed and was replaced by Apple. When I set up my computer again, the previous three years' worth of email messages downloaded into my Mail program across my 5 accounts. This meant that I was carrying a large number of unread messages in my mailboxes. A year later, this spring, my two largest accounts stopped downloading the messages from the server. Then last month, the next largest account did. Two days ago, the fourth account, which really was not that large, also stopped downloading. I still have one account that is.

This motivated me. i went through and cleared out all the mail except for a few messages overall and put them into the On My Mac folders. Then I tried again. The Mail Activity shows the program running through all the messages on the server, but none of them end up on MY computer. Then, with one account where I had completely emptied the mailbox, I deleted the account and then re-added it. Still nothing. I am stumped. Any ideas out there?