The Dropbox API does not block IP addresses or timeout requests intentionally. If/when the Dropbox API rejects a request, Dropbox will return an explicit error response.
I'm also not clear on what you mean when you say the request timed out, but you received a 499 status code. The status code would be in the response, but the error you included seems to show the connection failed so there wouldn't be an actual response from the server to contain a status code, and the output you shared doesn't show "499".
It looks like you may have truncated the error output though. Is there more to the error that you can share?
In any case, is there anything on your network connection, such as proxy, firewall, etc., that may be interfering with your connection to api.dropboxapi.com?