I won't say it's impossible, but it is very highly unlikely. Things have changed from the old days when you could take your chances like that. With the security requirements it makes it more difficult. Plus the ships seldom are not sold out before that last day. Ofcourse there could be a last minute cancellation.
It is virtually impossible since 911 to get a ticket at the port...first you cant even get close to the port if you dont show your ID and tickets or state your reason for being there. As Cruise Fanatic stated, most ships are pretty much sailing full. Now you can call an agent and try to get a last minute booking and have pier pick-up for your tickets. Might get a great last minute deal that way.