and the error message would be?

it says "login to server smtp-server.rochester.rr.com failed

https://support.mozilla.org/kb/cannot-send-messages

your provided deprecated the locality based server names years ago. My guess is they have finally turned them off.

Time Warner Cable list the following settings. I suggest you update your account settings

   POP settings:

       Account Type: POP or POP3
       Incoming Mail Server: mail.twc.com
       Incoming Server - POP3 port: 110
       Outgoing Mail Server (SMTP): mail.twc.com
       Outgoing Server - SMTP port: 587
       Inbound & Outbound User Name: Enter full email address
           johndoe@[division].rr.com
           johndoe@roadrunner.com
           johndoe@twc.com
       Inbound & Outbound Password: Email Password (same as Web Mail)
       SMTP Server Requires Authentication: Checked

   IMAP settings
       Account Type: IMAP
       Incoming Mail Server: mail.twc.com
       Incoming Server –IMAP port: 143
       Outgoing Mail Server (SMTP): mail.twc.com
       Outgoing Server – SMTP port: 587
       Inbound & Outbound User Name: Enter full email address
           johndoe@[division].rr.com
           johndoe@roadrunner.com
           johndoe@twc.com
       Inbound & Outbound Password: Email Password (same as Web Mail)
       SMTP Server Requires Authentication: Checked

If "SMTP Server Requires Authentication: Checked" is in Thunderbird SMTP Account Settings, I don't see it. If that setting is at the Spectrum (TWC) website, I don't know how to get to it. The Account Management at the latter seems to allow only for password (POP) change. Or maybe I'm misunderstanding. What was turned off, where, and by whom?

So it there any downside to the Thunderbird SMTP server Authentication method setting changed from "Password, transmitted insecurely" to "None"? Because, if done so, I have no problem in sending mail.

Lets look at what you just said here.

I know your email address. So ostensibly I can send email as you (or at least anyone using TWC as an internet provider) There is no password required to send mail using your email address on your providers server. So it there a downside.... yes there is. you email address is hardly yours if everyone can use it.

So to move back to your previous posting. Your provider is dumping servers other than mail.twc.com for receiving and sending mail. So have you changed the server names as required?

I have changed the SMTP server name to mail.twc.com after a TWC customer service rep told me about the new name. (I called them about my sending mail problem.) It made no difference. Their POP server now also has the exactly same name, but my Thunderbird POP setting is still set to the old name and I have no problem with receiving mail. I just tried to revert to my original Authentication method setting: Password transmitted insecurely, and got an instant (???) response that the connection timed out. I wasn't even asked tor the password. What I have zero recollection of is how did I come up originally with my SMTP password and where was it set so that the server looks for it. I went through all my Thunderbird settings and found no trace of that password. Nor do I see that setting anywhere on the provider side. Where can it be?