I believe what you described is a Gmail "feature." Emails from you to
yourself bypass your Inbox and do not download. They seem to think it
reduces "clutter" in your Inbox or something. (There are those of us who
disagree.)
If you have another email account somewhere, you can send a test email from
there, and it should appear.
This is annoying because those of us who participate in email forums (like
this one), never see our own emails come back to us, so we don't know for
sure if they went through ... unless we go online and login to Gmail. But I
prefer working off-line.
Andy