However insistent the client might be that the address exists, you can't argue with a message from the mail server responsible that says it doesn't. Mail server knows best, client ..... not so much!
There's always the possibility there's something along the way which is preventing delivery of the message. For example, our mail goes through our ISP and we have to enter into their portal all the valid email addresses we will accept mail for.
The email will hit our ISPs servers, which will check if the recipient is in the database of allowed recipients, and if not it bounces the message with a similar
No mailbox by that name type NDR.
There could also be a hardware device on the clients side (spam filtering, antivirus etc) which is bouncing the message with some obscure NDR message.
What I suspect is most likely however, is that the email address really doesn't exist.