I have this problem for years now. I was a happy user of the TB display (2011) connected to a Mac Mini (mid 2011) for about two years (years ago), then the problems started. Today the display will not start up at all with that Mac Mini. I bought a MBP about a year ago. Interestingly connecting to the MBP the display worked. For some random hours each time.
I tried everything written in this thread, changed cables, memory in the Mini, software updates, microcode updates. resets, found companies who could service this, which proposed to upgrade modules without promising to resolve the issue...
But I finally gave up. While the TB display is still great (for the minutes it works), it is not possible to use it for anything. I substituted the display with some Dell HD IPS monitors (three of these connected to the TB, HDMI and the Apple TV (wireless) ports of the Mini. The wireless connection of the third monitor through the Apple TV has some lag, but is perfect for reading documentation or reading the web.)
My educated guess is, some software optimization in the Mac OS introduced a few years ago by Apple and the growing internal heat (caused by the gradual collection of the dirt in the systems - inside the machine and also inside the display) causes the problem.
The problems with the Mac Mini and the TBD started quite suddenly after years of smooth behavior without problems. I just vaguely remember now, but I had some problems with the energy saver functionality after an OS upgrade. And then the display stopped to be reliable after that.
If someone could test this environment (I mean a Mac Mini and the TBD) with the Mac OS of 2011, without later OS upgrades or patches, I would be very eager to know the results.
Perhaps it is just the fan in the display (and in the Mac) which is turned on too late when some components are already too hot.
The internal heat in the Mac and the fan of the Mac I can regulate with smcFanControl, and I could easily make the problem come up earlier or later by changing the fan rotations. This problem is temperature dependent for me. This excess temperature is then probably transferred through the cable into the display. But there are other factors for sure, it is not that easy. E.g. somehow the display “remembers” the bad settings. Even after the system cools down, it will not start up again every time.
At early stages a power reset of the display helped. Then I thought the display is dead. Then I connected the MBP and the display worked flawlessly...
Each system is somewhat different. Some memory upgrades generate more heat, the TB cable is a very good conductor of the heat between the machine and the display. The software is very different in each machine, the heat production is very different. The last updates made the problem better (but only for the newer environments) but did not fully resolve it.
Our whole family is a happy user of the Apple infrastructure. We are using iMessage, FaceTime and all the other stuff all the time. We do not have any bad experience with Apple aside fro this display. But this is still bad and there should be a way out of such problems.