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.
Short answer, because they can. The airline makes their own schedule and can change it whenever they want. Departing and arriving early has more upside than downside.

Keep in mind, you must be at the gate *at the original Boarding Time*. Meaning, yes, if Boarding Time is 10:15, they can start boarding at 10:00 if they want and make last call at 10:16. If a passenger is not present at 10:16, they can close the flight and depart. This is rare, but possible. Point being, the only time that matters is Boarding Time, especially if the flight is delayed.\*\*

There are two notable exceptions to this:

1. Slot Controlled airports where the destination airport gives departure clearance. This is however relatively easy to get moved up.
2. Gate space at the destination.

\*\*In some cases, such as a remote stand, they load the bus once, at boarding time. So if everyone makes it through by 1016 and you're no where in sight. The bus, and your plane, are gone.