How do I fix Mail in Snow Leopard where I cannot send emails from my account to another account I've set up in Mail, but the reverse works fine?

Question

I have a problem with Mail in Snow Leopard in that I cannot set up two email accounts for myself and my wife with the same ISP and have them operate properly.  I've set up the accounts in strict accordance with my ISP's instructions for setting up email accounts on Mac Mail but I cannot send an email from my account to my wife's without a drop-down message appearing that Mail 'cannot send the email from this server.  Please edit the server list'.  No amount of tweaking appears to work.  Applecare consultants have so far been unable to get it to work and a recent visit to the local Genius Bar in the Apple Store was fruitless also.  My ISP's help line was unable to solve the problem even with a supervisor level consultant.  The ISP concluded it was some kind of 'bug' in Mail.
I could send emails from my wife's account to mine, but not vice versa; I could send emails to other addresees without any problem and I can send emails from my account to my wife's using the ISP's webmail.
Does anyone please have a solution to my problem?  At the moment we're operating one account (mine) on my Macbook and my wife's is set up on her iPad2.  As we prefer to use the Macbook for storage and (using Time Machine) for back-up of our data, the two accounts on separate machines is a nuisance.
Another factor is that I prefer to control when I download emails and set my preferences accordingly.  One of the so-called 'fixes' that were tried wound up downloading emails regardless of settings, even after removing my wife's account; the matter was remedied only after I had removed all accounts form Mail and started afresh from scratch.

Answer 1

You'll have to pardon my ignorance of some  computer terminology; I'm a retired manager and not a computer technician, hence my muddling up of client and server terminology.
Re your comment about Port 339: I did qualify this with (?).  As I wrote, it all happened at lightining speed and the guy at the Genius Bar didn't bother explaining what he was doing.  Port designation contained 3s and 9s to the best of my recollection.
That said, you seem to be spot on re setting up a 'dummy' client in Mail.  I set it up as you suggested and I can send to and receive from this client without a problem.  You appear to have identified the solution, notwithstanding the claims of the ISP's 'experts'.
The next problem I have now is sorting out my wife's mailbox that is so troublesome whenever I set it up on Mail.
All the other set-up advice to date yielded a similar result; inability to send from my account to her inbox.  Perhaps deleting her details from the account altogether and setting them up again from scratch may be the solution and I shall try that in due course.  The impediment that I have with this at the moment is that my ISP admits that it has 'a technical problem with its email servers' and some of its related web pages that prevent my doing this myself.  I will have to go through the telephone 'help' line and ask the technician to set it up at their end on my instructions.  This has some disadvantages in that I have to divulge the password to the account to the operator rather than keeping it anonymous by doing it on-line.   You may have solved the problem, but until I can successfully sort out a working mailbox for my wife i cannot definitively say so.
Many thanks for your perseverance.  You've made more sense than anyone else so far and that deserves some points, surely?