Balderdash!
You are having success with your iPad (you said?)
OK. I just tested one of my roadrunner accounts from Thunderbird
- sent to a YahooMail account
- sent to "self" = same account as sending from
- sent to another roadrunner account
ALL went to Sent folder with no alerts
ALL FAILED to arrive in TBird
I searched Spectrum.com for "mail"

Note: Your steps and screenshots may vary slightly based on your version of OS X.
- Open Apple Mail, located in your Applications folder or the dock at the bottom of your screen.
- Select Mail at the top of your screen, and then choose Preferences from the drop-down menu.
- Select Add Account to create a new account.
- Choose from the mail clients listed and then select Continue.
Note: To add your Spectrum email account, choose Add Other Mail Account... from the list.


- Enter the following information and then select Create.
- Full Name: The name that will appear in the From field of your emails.
- Email Address: Your complete Spectrum email address (including @charter.net).
- Password: The password for your Spectrum email account.
- You'll be prompted to configure your Spectrum email account manually. Use the information below to complete the required fields and finish the new email account setup.
Note: The following settings are required for setting up your Spectrum email on a laptop or desktop computer, smartphone, PDA, tablet, eReader or other mobile device.
- Account Type: IMAP
- Mail Server: mobile.charter.net
- Username: Full Spectrum email address (including @charter.net)
- Password: Spectrum email password (This field should automatically populate)
- SMTP Server: mobile.charter.net
- Choose Use only this server.
- Select Use authentication.
- Username: Full Spectrum email (including @charter.net)
- Password: Spectrum email password
See Apple's support site for more information about setting up and using different email accounts in Mail.
Top
- Username: This is your full Spectrum email address.
- Password: This is the password you use to sign into your Spectrum email account.
- SSL: This setting should be ON for both SMTP and IMAP.
- Protocol: IMAP
- Incoming Email Server: mobile.charter.net
- Port: 993
- Outgoing Email Server: mobile.charter.net
- Port: 587
- Requires Authentication: Yes, or checked
Note: Using both IMAP and POP simultaneously or using POP across multiple devices can cause error messages when trying to send, receive or access new email such as the following: "The mail server denied access to your account because another mail client was using it." To avoid these errors, we recommend using IMAP on all devices. Learn more about switching from POP to IMAP.
It seems to me that they may be giving you the cold shoulder because they have screwed the pooch and do not want to admit it