Maybe you need to change something to match a change your provider made. Successfully logging into your account on the webmail page just confirms that you know your username and password. The webmail and POP or IMAP servers are two completely different things.

Do you get any kind of error message?

Get a current list of server settings from your provider and compare those with what you have in Thunderbird to see if you need to change anything.

Can you post your settings here please?

https://support.mozilla.org/en-US/kb/ask-tb#w_how-to-ask-your-question

I receive an Alert that says the following: "The folder Inbox is full, and can't hold any more messages. To make room for more messages, delete any old or unwanted mail and compact the folder."

My Inbox has been cleaned out and old messages archived. I still receive the same response.

My settings are all fine and match up as they should. I tried to clear the passwords and restart as well, no change.

Have you compacted the Inbox? Deleting files merely hides them from sight; they may still be there occupying space in your file system.