I'm not entirely sure that I agree with what's been stated previously. Hear me out here.
I think that it is at least possible to have a scenario where too many plants that would otherwise NOT need CO2 will be limited by the CO2 found in the water. In a normal aquarium around 76 degrees, oxygen levels and CO2 levels are actually going to be very similar, around 8 ppm. However, low O2 saturation is not uncommon in non-planted aquariums, as still surfaces and overstocking can lead to less gas exchange and therefore begin to deplete the oxygen levels in solution. This problem is easily mended by increasing surface agitation and/or an airstone. This problem comes up very little in planted tanks because the plants also serve to add O2 to the water. Now, where does that leave us with plants and CO2? Like fish, plants will deplete the CO2 over time. However, if you were stocked in such a way that the CO2 would be removed very rapidly from the water (say, an entire tank of wisteria). It is entirely possible that the plants would use CO2 faster than it would re-enter the tank from the atmosphere.
Another interesting point: while CO2 and O2 have similar equilibrium levels in an aquarium, O2 has CO2 beat hands down in atmospheric levels. Whereas oxygen is about 21% of the atmosphere, CO2 is about .03%, or roughly 700x less. This suggest to me that CO2 would be much slower to re-equilibrate than O2, further lending to the idea that CO2 could be depleted in a moderate light but not CO2-obligatory situation.
I think that, more than anything, this is an interesting thought exercise, as I don't think that the OP is anywhere near these hypothetical plant levels, or even if they are feasible to reach. Interesting to think about though. I'm thinking about trying to carpet a tank with wisteria in the near future, so it might actually have some ramifications for me.