The only thing that may have physically gone wrong with my project is more than 3.3V being applied to one of the analog input pins.
That's definitely enough to destroy the chip through latch-up. One reason you must
never exceed Vcc+0.3V or go below -0.3V is to prevent CMOS latch-up. Wikipedia
will explain this.
Modern CMOS chips are fairly resistant to latchup and usually require high fault
currents to trigger it (100mA or so), but if you put the 5V rail on a pin that's more
than enough current to destroy any chip. Even fairly small value resistors on the inputs
can provide a lot of robustness against this kind of issue.