I am not able to send messages

Instead the following message appears:
16:30:47   MailClient.Storage.Application.ItemOperationException: Zprávu (Tělo zprávy nebylo přijato kvůli následující chybě: 
16:30:47     "5.7.1 You have exceeded the number of messages you can send
16:30:47 ")

This looks more like a provider message rather than one from eM Client.  Does your provider have limits?  Were you doing a mass mailing when this occurred?

Not at all, just a few messages. I do believe it is eM Client failure - I can send as many messages as I want when accessing the provider service directly. It is very disappointing!

The 5.7.1 is definitely a server error.  Your provider may have different rules for third party clients vs. their web client.  Also check to make sure your smtp settings are correct.

You are right, it has been some misunderstanding between the mail client and the server. What helped was to run diagnostics in eM Client Menu/Tools/Accounts - I like this program ever more. Thanks for your support, Jay!