If it says "mail sent" it sounds to me as if it sent it on its way. If you don't get a failed delivery response later then presumably it got to the destination; but you could always chase up and check it was received.
Sent mail is usually stored in your e-mail client in a folder, usually labelled "sent mail". Check to see if you have one. The e-mail client seems to be suggesting it could not store the sent e-mail in that folder. Maybe due to lack of space ? Maybe you need to do a bit of housekeeping, or check if you can assign more space to your mail in the options menu.