I'll need some more information to try to help you with this issue.
1) Do you have more that one account setup in your Outlook Express?
If so, can you successfully send a message to this recipient using the other account?
2) When you say, "...same receiver may have just been sent a previous e-mail seconds before..."
Was this successfull message sent using the same "DSL" account?
What I'm trying to determine is if the SMTP (Simple Mail Transfer Protocol, or the "Send" portion of mail server) is setup correctly in your DSL email account. If it isn't setup correctly, you should not be able to send any mail at all with this account.
If your SMTP is set correctly and you're able to send email from this DSL account, then my next question concerns the message recipient.
3) Reference your comment. "Since then I often get error messages saying an e-mail I'm sending cannot go to someone..."
Do these messages that return an error message go to the same recipient, or are the addresses all to the same Mail Server?
The "Mail Server" can be identified by the words after the "@" symbol in the email address. @hotmail.com means the message is going to the Hotmail mail server.
If all the returned messages are going to same mail server, then my guess would be an error with your setup with that specific mail server.
If the messages are going to a Compuserve IMAP server account,
this article should help.
Finally, if nothing here helps, can you post again with the mail server name of your DSL account (not your email address, only the information following the "@" symbol). Please include the mail server information of the recipient's address, also. If you've seen this same error to different mail server's, please include the server information for those, too. What I'm trying to determine with this information is if there is a common denominator to the mail servers where this problem occurs.
I hope something here will help.

I hope yours is a great day!