Neopixel led strip stops responding after change in brightness
It is almost certainly a power issue. The Adafruit page about one of those types of Neopixels says:


  Don't forget, with 256 LEDs, you could use over 15A of current if you turn on all the LEDs on to white (which we really do not recommend because we don't think the flex PCB can handle that much current). Try to keep the current draw at under 5A ...


Increasing the brightness increases the current drawn. Do you have a heavy-duty power supply? Are the connecting leads sufficient for (say) 8 amps of current?



New information based on comments made

Based on comments by Lorenz I now think it is more likely to be that they are overheating.


  I have the strip embedded in a picture frame with epoxy resin over it. 


That makes it difficult for the ICs to cool down, and LEDs generate heat. If you search for "neopixels heat" you find quite a few matches. Some suggestions made are to use a heat sink or a fan. If this isn't practical for you, then running at a lower PWM rate (ie. not as bright) and not as saturated colour, could help keep the heat down.

You don't want to "cook" one of the early pixels, because each one has to pass the data on to the next one in sequence, so if you lose an early one, you lose everything after it, particularly if you can't get at it to replace one.



Postscript regarding heat

I have been testing a NeoPixel with a temperature sensor. Displaying full white (255/255/255) it has reached about 44°C. So clearly it warms up. However this is in the open air with just some sticky-tape holding the sensor onto the LED. According to the spec the chip can operate at up to 80°C, so this is within range. 

However if the whole thing was sealed in epoxy, then the heat being generated would have much less chance of escaping into the ambient air, and thus would be likely to get a lot hotter.