Pretty closely. The 2:30 arrival time is something recently being enforced by the Dept of Homeland Security. They want everyone on the ship an hour before it sails. You used to be able to arrive later but not anymore. I have not arrived late for a cruise since the new rules, so I can't tell you how rigid that time frame is.
If you can't gaurantee you are at the ship before 2:30, then I suggest planning on arriving the day before and spending the night at a hotel. The ship gets charged port fees based on time docked, so in order not to be charged additional fees they need to leave on time. There is a little bit of "slop" built into that schedule but not much.
Carnival Cruise Lines:
Fantasy 90, Celebration 95, Victory 05, Conquest 05, Spirit 6/06, Freedom 6/08, Holiday 7/08, Victory 6/09, Valor 9/09, Dream 6/10, Splendor 8/10,Conquest 8/11,Magic 11/11,Triumph 4/12,Spirit 7/12,Freedom 11/12, Triumph 2/13 - The fire cruise
NCL Dream October 2006, RCCL Mariner OTS June 2007, Celebrity Silhouette 2013