OK so how about if you have a situation where you have +/- voltages created by a half wave rectifier with the 16V filter caps seeing about 3V over their voltage rating and everything sharing the same ground?
First off, don't do that unless you're already stuck with it and can't change, and for reasons unrelated to ground planes. Rectifier/filters need special wiring on the grounds, and ground planes are not the way to do it. Rectifier/filters have relatively huge pulses of current on the nominally ground side, and this stuff simply must be on its own wires from transformer to rectifier to filter caps to avoid hum issues. In addition, these pulses can make the wiring to them ring at RF, and a ground plane setup may in fact carry the RF to sensitive places. Half-wave makes this worse because the pulses are less frequent, and therefore higher amplitude. And filter caps should never be operated at over their voltage rating unless you really have no options. If at all possible, trade those out for the next higher voltage rating.
If you're stuck with it, there may be ways to cope.
Can you get a ripple/oscillation on the ground that would cause certain types of IC chips to malfunction? And would a ground plane make that worse?
All that being said, causing certain types of ICs to malfunction would be unusual just from the ground plane; but possible, as a ground plane is, among other things, a tuned high frequency transmission line to everywhere in the circuit. The tuning and other issues get critical, as does the specific IC. But I think that other issues are your problem instead of there just being a ground plane. Details always matter, though.
