Hi Richard, if you had a Google account using the CableOne email address as its name, it had no connection to your CableOne emails.
If you added Gmail to that account, the Gmail address becomes the primary name of the google account, but you can still sign in using the CableOne email address and it still has no access to your CableOne emails.
If you chose to import your CableOne messages and/or contacts into Gmail using Import from another address under Settings > Accounts and Import, then your CableOne messages are copied into Gmail and should still be in your CableOne account unless you selected the option to delete them as they were copied.
To access your CableOne messages, please sign into the CableOne account at their webmail service, not at mail.google.com.