After an exchange of emails that was courteous and helpful, Sony recommended I perform a System Restore. Yuck. That would blow everything out. So, I had the genius idea of using Mozilla rather than Netscape. Since I haven't been using Mozilla, it's an older version, and might be not be prone to the problems that Netscape has been having.
The good news: It downloaded my mail.
The bad news: It toasted the mail it downloaded. I think all three hundred pieces went bye-bye in one way or another. Text is interchanged or absent, headers are all messed up...it's not pretty. I don't think any of it is salvageable...or the bits that are won't be worth looking through to save.
The okay news: I've downloaded one mail since then, and it came through okay. Something simply must have gone wrong with the enormous quantity of mail that it was trying to deal with. *sigh*
It could have been worse, I guess.