I own an NEX-5N and as others have reported, I have experienced auto shut-off and thermal shut-off. I had read of others using heat sinks to cool the unit, and I thought I might make a custom one for this purpose. On advice of friend, I bought a surface temperature reader (About $90 from Vernier).
From it I got some interesting profiles. But to read the chart you need to know what "Label ON" and "Label OFF" mean. Pictured below is the NEX-5N with the LCD open to reveal the back with a label that I've already begun to pry up with an X=acto knife.
And surprise, there is a carefully designed heat sink already in place.
And now the data. Click on the chart below to see a larger view.
First, I only tested one camera; mine. Your mileage may vary.
The steep drops are characteristic of thermal shutdown which turns the camera off.
The slightly shallower drop is characteristic of timer shutdown which returns the camera to idle.
The jittery lines are characteristic of having the LCD open. For these tests, LCD open means the LCD as far vertically over the camera as possible with the screen remaining parallel to the back plane of the camera.
Clearly the combination of removing the label and opening up the LCD greatly reduces the thermal load. (How much open the LCD has to be I will investigate next.)
However, this is not sufficient to keep the camera from turning off. I believe the reason for this is the 30 minute Power Saver feature which has a variety of settings up to 30 minutes, but no "never" setting.
As the lowest line on the chart shows (Idle, Label Off, LCD Closed) the off time can be extended in idle mode by accessing the MENU (as I did at about the 13 minute mark).
I have tried to find a way to do this in video recording mode, but without success. MENU is not an available function and adjusting the ISO or the Exposure Compensation did not reset the Power Saver timer.
Of note, even in the last case, the thermal warning indicator did come on at around the 25 minute mark.
Obviously Sony knew this was an issue. Otherwise why would they have built in thermocouples, warning indicators, automated shutdown sequences and punch 150 plus holes in the back of the camera?
The combination of the label and the full closure of the LCD (of lack of closed venting) is an issue.
If my analysis is correct, Sony should add a "never" setting to the Power Saver mode.