Can I trust Time Warner Cable?

I clicked on "Helpful", not Solved, in the email I received about this. Although it may be "solved," it's a little early to tell. I had called TWC before asking the community. Called 1-800-892-4357, the # on their webpage. The girl who answered could not tell me why I was receiving the warning message. She said that I should be fine to receive their webmail. She claimed she wanted to "take over" my computer so she could see the message, but then claimed she "timed out" waiting for the app to download. (Syrupy slow internet-- another of the suspicious things going on) Instead of answering why I was receiving the warning, she forwarded my call to some fast talking guy who started rattling off a lot of numbers in some "Verification Process". When I didn't get all the numbers written down correctly, I called the phone number he claimed was their call back number. It was the "Psychic Source"!

So I called TWC again today with the 800 number you gave and talked to a guy who was actually helpful instead of suspicious. He had me clear the History of my web browser and the cache. Then he checked the speed of their service to my modem (it's full speed). He said that he suspects my 2012 Airport Extreme router is bad already. That might explain the sketchy wifi around the house, but it still doesn't satisfy my original question about why I was receiving that warning message, (have not seen another one since today when we cleared the browser history), or relieve any fears of intrusion into my iMac.

It's baacck! After working with someone at TWC who tried to be helpful (without answering the question), that warning message is back in Mac Mail. It's titled: Verify Certificate. The identity of "mail.twc.com" cannot be verified. The certificate for the server is invalid. You might be connecting to a server that is pretending to be "mail.twc.com" which could put your confidential information at risk. Do you want to connect to the server anyway?

Is it a coincidence that the bank called recently that someone tried making a fraudulent charge to our card, so they had us close that account?

I started receiving robo calls every day from pfishers.

Thanks for the support information about "an SSL certificate that's signed by an unknown authority (such as a self-signed certificate)". It was interesting that Apple doesn't seem too worried about it. I still have connectivity issues that TWC is blaming on my Airport Extreme router, but I won't suspect their internet service then. Thank you ChitlinsCC.

I'm over talking to TWC. I called Apple and their Support Experts helped me. The first one shared that some Cable companies have been making changes to their systems. She knew which settings we could change in my Airport Extreme router to improve connection. She went into Internet Options and Configure IPv6. The second Support Expert said that we should be able to trust the big ISP companies. He advised that I could update my operating system to the latest version for the greatest security, or I could uncheck "use only SSL" in my mail settings, or check "always trust..." in the warning message. The warning message should go away and I hope my router settings improve connectivity. Hopefully, I get to keep this Airport Extreme router for awhile. Thank you very much for your assistance, ChitlinsCC