emails not delivered & not bouncing back

Is this isolated to one user on your net or is it all users having this problem.

To answer your question of no bounce back, Yes it is possible to send an email from a nonrecognized domain and not receive a bounce back [the senders email addressis not recognized by any name server therefore mail servers processing the email can not send a "bounce back" to the sender.

The problem sounds like a MX record for your domain is missing from the name server[s] your using.

I've seen this happen and it works like sending mail into a black hole... nor indication on either side unless the two people speak with each other and talk about the problem.

Unfortunately it's usually a blocked domain or firewall issue on oneside or the other (or in your case, maybe in the middle). The fact that hotmail e-mails go through, means that both sides need to check their blocks/ports etc and make sure the other domain is not in the list.

What are you using for a firewall?Do you allow relay? Another thing to check is if either your site or theirs is on a block-list because of relaying.

Please note: since you do not have your own mail service [own mail servers] rather use a 3rd party POP3 mail service, there isn't any "FIX" you can do. The problem is with your mail service, not within your network.

As your question asked "...user who is sending emails and has confirmed they are not getting delivered and they do not bounce back. Is this possible?" the answer is yes it is. As Anna pointed out, there is an email "black hole" or "never never land" where undeliverable email such as the type of service your using ends up [Purgatory more like it] and neither the sender nor receiver know about it until much later. [the email service provider doesn't provide NDR's to users. The nondeliverable email is discarded before it even reaches the mail server [possibly a filter put in place by the mail service]. I have filters on our Exchange servers that drops all email from domains that are known spam services or are blacklisted mail services. Those emails don't even reach the mail server [re they go to the black hole of email]. No Bounce back notice, no Non Deliverable Report is ever generated. The mail is simply discarded. This sounds like what is happening with your POP3 mail service.

One thing you could try:

See if you can send (from your e-mail account at this location) to the exact same address that she sends to. Call the person and see if they get anything.

If not, there's nothing you can do.
If they do get yourtest mail, then there's some block or filtering going on somewhere ... at that point I'd probably resetup her Outlook Client and verify her account/profile and maybe even change her e-mail address (or add an alias if the other company permits aliases).