Thank you, CT, for getting back to me on this.
In mail preferences under General > 'Downloads folder:', Desktop is selected.
In the folder ~/Library/Mail Downloads I find the 2 PDFs I sent to myself, but none of the pictures I sent, or the important attachment (a small app) that was sent to me from outside.
In the list of incoming e-mails, all e-mails that were supposed to contain attachments are marked by paperclips. But some, the PDFs, show up in the body of the e-mail, whereas others do not, especially I notice the absence of any pictures. Only the PDFs appear in the library mail download folder.
I notice that when I send myself attachments whether pictures or PDFs the paperclip appears in the received mail but not in the Sent list. Mean anything?
This sure looks like a bug. I just a few days ago upgraded to Lion 10.7.4. Everything else works beautifully. Unfortunately, I can't send myself experimental attachments from anywhere else except my own computer, so my experiments are limited.