WordPress cron (which Postie relies on) doesn’t run unless a page is accessed on the site. So if you send an email, but nobody accesses the site for 3 days Postie won’t be given the chance to fetch the email and publish the post.
To ensure that Postie runs smoothly on a low or no volume site you need to ensure that a page gets hit (any page is fine). Use something like cron + curl on Linux or install curl on Windows and use the Scheduled Tasks control panel. See this page for more info.
It is also possible to turn the WordPress cron off. Please make sure something like
is not in your wp-config.php file.