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 or use one of the online cron services. See this page for more info.
It has been reported that enabling the "alternate" cron has worked for some people.
In your wp-config.php file add:
define('DISABLE_WP_CRON', true);
define('ALTERNATE_WP_CRON', true);
add this before the line that says “That’s all, stop editing! Happy blogging.”