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You need to do a real submission in order to properly test, the "Test button" will sent the notification to your account email address. Please check this guide: https://www.jotform.com/help/314-How-to-Properly-Change-and-Test-a-New-Email-Recipient-Address
In addition to this, you may check if the recipients are in bounce list: https://www.jotform.com/help/262-How-to-Remove-Your-Email-Address-from-Bounce-List
We checked with network solutions and looked to see if there was a IP block for Jotform. We did not find a block nor did we see Jotform trying to make a connection with Network solutions mailserver ip 126.96.36.199.
As for testing we did remove the email and added a Gmail email. That email worked fine. The issue seems to be from Jotform not sending the notification to Network solutions. We also experienced this same problem with a @att.net email address.
How shall we proceed with trouble shooting?
Your recipient email was in bounce list:
It has been white-listed now, so this should solve the problem. I would recommend you to white-list our Jotmail IP addresses within your email server (or ask for it to your email service provider).
Read this article for more details: http://www.jotform.com/help/145-Whitelisting-JotMails-IP-Addresses
Let us know if you need more help.