Yep, in fact this is especially noticeable in situations when your GMail is not working for whatever reason. Once, I had my GMail account temporarily disabled for a day (I set up some filter that immediately deleted and forwarded certain e-mails which apparently got triggered too many times and caused a temp shutdown) but was still able to send e-mails to myself.

Lesson here is, if you use GMail, don't e-mail yourself to check if sending works at the moment.