IMAP synching issues on OS X Mavericks
Up front I have to say that it is known factor that mavericks has (had) some issues with IMAP mail providers. There was an upgrade that supposed to address that. 

All in all, make sure you have the correct IMAP settings at your provider.

Here is an example for Gmail.

Starting from the settings in your web mail client (ie Gmail)

Go to your web mail (gmail) and in settings, check all folders (Google calls that Labels) you want to see on your computer (show in IMAP).

Just to make sure:


Disable junk mail filtering in your Mail.app on your computer
Disable all Rules in your Mail.app on your computer.
Also take a look for new mails in the Archive folder, which is an equivalent to the Googles "All Mail" folder.


In Mail.app under Window--click on the activity and see it work once you click the Get Mail. It will show you what is going on.

Here are recommended setting for Apple mail from Gmail: