I posted this a week ago, it might help you out. Worked for me!
Mine started doing this after QB forgot my email info. I had previously had mine setup manually, but used the auto setup with google this time ( which was laborious) After setting it up it would only send 3-7 emails and then give the network connection failure message.
After fiddling around a bit, I decided to just set it up again manually, saying "others" on pulldown menu when it asks what email provider. The outgoing SMTP server for Gmail is smtp.gmail.com then I clicked the box at bottom SSL/TLS and entered 465. It works fine again. If you have a different provider you can look up the SMTP server info for them individually.
If you need more info on how I did this, feel free to ask. Hope that helps someone with the same problem!
