Multiple Ethernet Clients With One Ethernet Shield?

Is it possible to have more than one EthernetClient/open connection on a single Ethernet shield?

Basically what I'm attempting to do is have one connection to a server remain open to do long polling and have a second connection open doing normal HTTP posts.

Something like:

void loop()
{
  if (client1.available()) {
    //read out client1 and act on long poll response
  }

  if (!client1.connected()) {
    //reconnect long polling
    client1.println("GET /notify HTTP/1.0");
    client1.println();
  }

  if (client2.available()) {
    //read out client2 and act
  }

  if(time_to_update < millis()){
    //send normal post on client 2
    client2.println("POST /api HTTP/1.0");
    client2.println();
    //...

    time_to_update += 60000;
  }
}

I know I could always just disconnect from the long polling server, but if I could keep the connection always open I feel like it would be less complicated and quicker to respond.

The documentation for the standard Ethernet shield says:

It supports up to four simultaneous socket connections.