I use the Teensy 3.5 on a carrier board. The carrier board has a 3V3 voltage regulator, so I connected the 3V3 pins from the T3.5 to the 3V3 from the carrier board directly.
Between the LM1117SX-3.3 on the carrier board and the T3.5 there are 2 capacitors: 10uF and 100nF. If I power the board, the 3V3 voltage needs around 5ms to rise from 0 to 3V3.

Now I have the following problem: If I connect the power supply from the carrier board, the T3.5 won't boot. To test this I turn on a LED in the setup routine in my Arduino Code.
If I disconnect the power and reconnect it after a short duration (< 3 seconds), the Teensy will boot without problems.
If I disconnect the power and reconnect it after a longer duration (> 10 seconds), the Teensy won't boot.
If I connect the USB cable directly to the T3.5, the Teensy will boot always.
If the Teensy is powered from the carrier board but didn't boot (after a long reconnect pause) and I plug in the USB cable, it won't boot as well.
If I press the Reset button on the Teensy while it's running, it will never boot. I need to reset the power to achieve a boot again.

The power ramps on the regulated 3V3 supply voltage look always the same, no matter if the Teensy actually boots or not. Always a 5ms ramp.

I saw this thread: https://forum.pjrc.com/threads/44704...h-three-T3-6s? and tried to solve my problem with a 10nF capacitor between the Reset pin and GND.
It worked for around 3 tries, then I pressed the Reset button. Now it won't boot, no matter what I do.
After enabling power, the voltage on the reset pin is around 0.1V. If I press the Reset button, it jumps to 3.3V but I have the same problem as stated above: It won't boot after pressing the Reset button.
I removed the 10nF cap on reset, but the problem is still present.

What could be the reasons, that my T3.5 won't boot with the external power supply?
Why does the time between resetting the power supply matter?
Is there a way to revive my "dead" T3.5?

I have a total of 9 such boards with a T3.5, all have the same problem. On only one I tested the reset capacitor, with mediocre success.
At the time of powering the Teensy, there are no relevant power sinks connected (all pins are inputs).