Is it possible to run multiple TeamSpeak 3 client applications?

You can indeed. TeamSpeak 3 supports connecting to multiple TeamSpeak 3 servers at once via its native "tabs" (similar to tabs on your internet browser), however if you want to start multiple TeamSpeak 3 clients, simply launch the TeamSpeak 3 client with the "-nosingleinstance" commandline parameter. Keep in mind that any client launched after the first one will not log any chats.