We have tested that board with voltages as low as 3V, and we are not really sure how it could fail in a way that lets it work at 5V but not at 3.3V. That makes it especially strange that you are seeing this behavior from two units. Can you try measuring the on-board regulator voltage when the board is powered from both 5V and 3.3V? You can do that by touching one of these points with your positive multimeter lead and GND with the negative lead.
If you have an oscilloscope, could you use that to look at what is going as well? Specifically, please check for unusual noise or transients on the supply voltage and the regulator voltage.
Also, I suggest making sure your GND wire is seated securely in the breadboard. In your picture it looks like the it might not be making good contact (though it might just be a long pin instead).
- Patrick