Question: It happened to me just one time, I don't know how common it is. I wonder why it can happen.

I was hanging at duty-free shops before a flight from Germany to Turkey. I heard the *last call* announcement 40 minutes before the departure time and my name was explicitly mentioned. When I went to the gate, boarding had been completed except me. Normally that airline closes the gate 20 minutes before the departure.

The pilot announced that he was talking to the control tower to take off earlier than planned. At the end we couldn't take off earlier, but I didn't understand why they tried to do this.

Answer: 
My retort would be "Why wouldn't they try to do this?". Nobody is likely to be unhappy about arriving earlier than expected. I doubt anybody particularly enjoys waiting in an airport departure lounge and would complain about having their time cut short. So why not try to get everybody on the plane early, depart early and get to your destination early? If nothing else it means a slightly shorter work day for the flight crew.

Sometimes this is caused by unexpected [slot restrictions](http://www.eurocontrol.int/articles/atfm-slots) at the destination due to oncoming weather. In such situations they calculate the cost vs benefits of an early departure. In this specific case only 1 already-checked-in passenger was missing and was easy to trace. 

My assumption would be that the last call didn't mean they were about to refuse boarding, just that they only had a few passengers left to board and wanted to try to get them on board. If you'd taken 20 minutes to get to the gate they'd probably have still let you on, but don't count on it and if there is ever an announcement related to your flight don't ignore it just because you still have plenty of time before departure.

I should note that none of this is from experience on the airline side of things, just my experience as a traveller so there are likely more informed answers to be had.
This could also be related to air crew duty hours.

Duty hours are limited to ensure the crew is getting enough rest to work safely . The hours logged include time spent sitting in the airport waiting for the departure time or on standby waiting to be assigned to a flight. If part of the crew is near their duty time limit, departing early might prevent calling in a standby person to allow completing the flight legally.

If it's near the end of a month in which there have been a lot of weather delays, the airline company may be in a situation that there is no one remaining to call in and their next choice is cancelling flights because they can't legally start them. If they can get the flight off early enough to land within the limited duty hours then they don't have to cancel a later flight.