We have been very lucky with our gear until now and haven't experienced any problems ever. However, luck is bound to run out at some time and if you use gear a lot, something will break at some point.
The statistics show that the shutter of a Canon 6D usually dies at about 113.000 exposures. And now the shutter of one of our cameras broke. (We don't know the shutter count right now as the camera still hasn't come back from the repair centre)
How Does a Shutter Work
Every camera has shutter curtains that are closed by default and protect the sensor from light. When you take a photo (aka releasing the shutter), the shutter curtains open to let light fall onto the sensor for a certain amount of time and then close again. Depending on the type of camera, there might be a mirror in front of the shutter that will fold up first (in DSLRs) or a mirror that is transluscent and will let some light through (in DSLTs). This article has some good animations.
Here is a video of the exposure process of a DSLR in slow motion:
When the Shutter Breaks - Symptoms
As you can see in the video, a DSLR's exposure works by mechanically moving parts around inside the camera body. Mechanically moving parts are bound to break at some point.
When it happened to me, I first realized that my shutter started to sound differently. However, at the time I dismissed it as me just being paranoid. A little bit later, I noticed that all of my photos had a bright reflection in the top right corner, and it wasn't lens flare.
Here is an example photo, taken of a white wall in even light:
At a fast shutter speed, there would also be a dark shadow in the left corner as the different parts don't move fast enough to get out of the frame:
The error showed up very suddenly in between two shots that were taken within split seconds of each other. Below is a crop of the far right corner of the photos in portrait format (left before shutter broke, right after):
This video shows how a broken shutter looks - I bet mine wasn't very different (I should totally ask the repair centre to keep the broken shutter - as a memory!):
Fixing a Broken Shutter
I sent the camera to a repair center. Even though it costs some coins, it is not that expensive than one would consider to buy a whole new camera instead - at least if you have a Canon 6D like us.