This problem is really stumping me. I made a simple printed circuit board with an ATtiny85, a 128x64 I2C OLED display, and a Sharp IR analog distance sensor. You can view the circuit here: circuits.io.
If I turn the system on using the simple 3-pin slide switch that I put on the board, the screen does not turn on. I can see that the system is powered though, because I can see a little bit of red light escaping from the IR sensor where some of the IR light borders on the visible end of the spectrum. If I turn the board off then immediately turn it back on, the display powers up just fine. If I upload a new program to the board and use the programmer's 3.3v power I don't need to do it twice - the screen works first time.
I'm using a 3.7v LiPo battery, so I know power is not an issue (everything is 3-5v tolerant, and the LiPo battery can deliver plenty of current). Why do I need to turn it on twice in order for the screen to work? Thanks!