I am new to OS X. Yesterday I set up my email accounts in Mail. It immediately synchronised my emails from my inbox. However, since then I don't get my mails, although I see them through webmail.
I heard Mail in Mavericks had problems with IMAP, so I tried to install Thunderbird and Postbox as well with the same result which means I can see the emails I received before yesterday, but none of the new ones.
I even tried rebooting but that did not help either.

What is the problem and how can I solve it?

UPDATE:
I do *not* use gmail, my account is hosted by a private provider.