AVR (attiny24) resets itself driving more than 5 outputs under no load
The problem is that there is a spike when turning on all the pins, even though they are not loaded, and this sapped enough power to cause a brown out.

The Atmel design notes specify the cap should be as close to the pins as possible and it worked when I put a 100nF cap directly on the chip's VCC and GND on the breadboard.  There wasn't a huge gap between the chip and my power rails, just over an inch, so this shows how important it is place the cap correctly.

I've used a lot of AVRs and this one seems particularly unforgiving in this aspect, especially as the outputs were not drawing any current externally.