That's true of any solvents initially trapped in the plastic, but different plastics go through different degradation processes, many of which create fresh volatiles. Celluloid and [cellulose] acetate are especially notorious because their degradation processes are autocatalytic and produce fairly aggressive volatiles that catalyze not only further degradation of the plastic but also of things like paper and lungs. I don't think you'd identify that as a "plastic smell" though.

Separately, plasticizers incorporated in many plastics ("new car smell") often continue evaporating at smellable levels for decades.