Introducing SwaggerSocket: A REST over WebSocket Protocol
blog.wordnik.com
blog.wordnik.com
To the extent that you can't send multiple requests without getting answers, that's your fault for using synchronous requests, not for using not-a-websocket.
WebSockets aren't performance magic. They're slightly glorified TCP sockets. If you've already got a TCP protocol (like, say, HTTP-1.1), it may be convenient to put a WebSocket interface around it, but there's no other advantage. And this is a particularly weak case since if we support websockets in the first place we almost certainly have full HTTP1.1 pipelining support.
fewer open connections while awaiting results, avoiding browser limitations
Think of desktop apps that keep a connection to the server and are awaiting messages from it. I think it might get interesting, maybe not the best application of WebSockets, but interesting. Your Email client wouldn't have to poll every X seconds to see if new mail has arrived.