Well, this will depend on your email server (pop? imap? other?) being up. Is it local on your machine, or external? Either way, being able to launch an email client (or some "new mail" notification app) shouldn't be a problem. It will simply not notice new mail until the mail service comes up - but it won't fail. It'll be as if the mail arrived a little later.
Sure it can fail, when you start it up it'll most likely fail to login to the mail server, whether it's local or not, if certain services aren't already started. If you're using local direct access via maildir or mbox, that'll work fine but most people access remote server for their mail.
Helge Hafting
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