Hi Daniel:
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I am really glad you told me about adding your second sentence, because I thought I was losing my mind.

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We did not, on the problem boards in question, program the bootloader ourselves, but rather, purchased assembled/programmed from Sparkfun. (Since then we have also played a bit with bootloader programming though, but on other boards. I certainly agree though that when we do program ourselves, we could easily get the fuse settings wrong....)
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I will look at various things, including power supply voltage coming in to the Arduino, with the scope.
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Thanks for letting me know your feeling about things and what might (or is unlikely to be) the problem - that is very useful.
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To clarify your thoughts, are you saying that:
a) It is unlikely the chip has lost its programming, period? (Which is odd since reprogramming fixes at least some manifestations of the problem....)
or
b) If it has lost its programming, it was not spontaneous but rather, due to some circuit design error on our end? -
If you have any other thoughts about what I might check, of course please shoot them my way. And if I have any further info to share (either for my benefit by asking you questions, or for other's benefit in the future), I certainly will.
Regards,
David