As we do not currently have any open issues regarding messages, I would advise you try our basic troubleshooting steps. If they are not successful, I can reach out to our Technical Support team for you.
1: What browser are you using? Upwork works best in Chrome, but we still advise, if you are having an issue, to test the same action in another browser.
2: Clear your cookies and cache
3: Follow the other troubleshooting steps in this article: https://support.upwork.com/hc/en-us/articles/211067758-Upwork-Messages-Connection-Troubleshooting