said by aurgathor:
Of course in my case the root cause was an improperly designed circuit that required 3+ volts to function properly instead of going down to 2.6 - 2.7V.
Typical "3V" CMOS chips are actually designed to accept VCC = 3.3V +/- 0.3V. So the lower end of acceptable VCC is actually 3V which may be higher than a partially discharged CR2032 battery.
Now, the current drawn by all CMOS chips increase with lower temperature, so VCC could go down even more than acceptable range.
Solution is a buck-boost DC-DC converter IC to supply stable 3.3V from a single CR2032 cell. But that of course will cost an extra buck or two.