E-mails are sent (!), but CUSTOMER SEES ERRORS - "an error occurred while sending message"

[Jo!]

So, this is the issue.
E-mails are sent - messages reach me in my postpox.
Either - TEST E-MAIL, or MAILALERT 1.1. (2.2 not working), or CUSTOMER SUPPORT - all e-mails are OK,
BUT
in Front Office customer sees errors after sending e-mails,
in BO - also there's error indicated when test E-mail (from PREFERENCES- EMAIL - SEND TEST EMAIL).

AGAIN - e-mails are SENT!!!