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: 
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.
I run into this quite frequently on the YQR-YYZ (Regina, Saskatchewan-Toronto Pearson, Ontario) route. My strong suspicion is that it's due to frequent delays in Toronto. Sometimes due to congestion or weather, you may not be able to land when you expect and can be delayed in the air while awaiting clearance. Also, the taxiing can sometimes take awhile depending on which runway you use to land, so wind direction and congestion (which affect runway choice) can elongate the wheels-on-the-ground-to-plane-at-the-gate time.

Of course, sometimes wind conditions can slow you down en route, too.

The sooner you're in the air, the more likely you can be at the gate at the scheduled time, irrespective of what's going on at the destination, and when the destination is a busy airport, that's important. (When the departure airport is a quieter one, it seems a lot easier to get into the air early. I've been in the air as much as 20 minutes early flying out of YQR, on jets as large as an Airbus 319.)

So... passengers should expect it. The departure time is an estimated time, but it can change out of circumstance, necessity, and convenience. And of course, as mentioned by others, arriving at the destination early is never a bad thing, especially if you have a meeting to attend or a connecting flight to board. The worst case scenario is that you're stuck in the destination airport longer and the originating one shorter.