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.
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.