While you may not have a firewall on your machine, you might have a firewall blocking access to your remote database. Ensure that you are able to bypass the firewall if it is in place.
The first thing to do is to verify that you can ping the database server from your client machine. If you cannot, then the problem is a network problem, not an Oracle problem. If ping is successful, then verify that you can tnsping the database listener. It may be that you have a blocked port. If tnsping works, then you should be able to connect to the remote database just fine. If you still cannot connect with Toad, try connecting with SQL*Plus instead. You may find another error is your cause that Toad is not reporting.
Finally, the TNS-12535 error is being tripped because a timeout is being reached. If you can tnsping your database listener, then the timeout is in creating the connection to the database. In your database server's listener.ora configuration file, either increase the value for your CONNECT_TIMEOUT parameter or set it to zero so that connections wait indefinitely.
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