This happens to me even when the credentials/authentication method aren't changed, i.e. I'm simply making a change to the connection string to point to a different server, etc. I have dealt with this issue quite a bit and the following workaround works for me:
1) Make whatever changes are necessary, then select auth method "Windows Integrated Security" or "Credentials Not Required" and click Save.
2) After changes are saved successfully, go back and change auth method to "Credentials stored securely in the report server", enter credentials, and then click save. It should be successful this time.