I was taking a photo of a nuclear radiation counter with my iPhone, and the LEDs weren't displaying properly in the first photo, even though it looked "solid" and not flickering to me.
Was this caused by the LEDs having a similar cycle when flickering as my iPhone?
Almost certainly those LEDs are flickering. There are valid reasons why LED displays are designed so that only some of the LEDs are on at any one time, but I won't go into the electronics here. Apparently your shutter speed was faster then the total LED refresh interval, so the camera only caught some of the LEDs on.
Use a slower shutter speed, like 1/50 second, and you will probably see the LEDs as intended in the picture. Some may still seem brighter than others due to having been at the right time within the refresh cycles. You need about 1/4 second or slower so that there are enough refresh cycles in the exposure that plus or minus 1 won't make any difference.
