The negative capacitance operation of a ferroelectric material is not only an intriguing materials science topic, but also a property with important technological applications in nanoscale electronic devices. Despite growing interest for possible applications, the very existence of negative capacitance is still actively debated, even because experimental results for ferroelectric capacitors with or without a metal interlayer led to quite contradicting indications. Here we present a comprehensive analysis of NC operation in ferroelectric capacitors and provide new insights into the discrepancies observed in experiments. Our models duly account for the three-dimensional nature of the problem and show a good agreement with several aspects of recent experiments. Our results also demonstrate that traps at the ferroelectric-dielectric interface play an important role in the feasibility of stable negative capacitance operation in ferroelectric capacitors.