Was on the Dream last month. Here are my observations:
We arrived at 10:45, so about 15 minutes after the port supposedly opened. There was a LONG line to get through the first check-in point (passport check), then a relatively short line to go through security, and a pretty short line formally to check in. Having said that, we were given boarding number 15 when we got through security, and it took almost an hour after boarding officially started to be allowed to board. So, from when we got to port to when we boarded, about 2 hours, maybe a little less.
When we left the ship, we got through customs at about 9:30, and the line for people waiting for the port to open for the next set of passengers was formed and not short. So I think if you really want to get on board early, you need to get to the port REALLY early and be willing to stand there and wait.