Precise dissection of cells with ultrashort laser pulses requires a clear understanding of how the onset and extent of ablation (i.e., the removal of material) depends on pulse energy. We carried out a systematic study of the energy dependence of the plasma-mediated ablation of fluorescently-labeled subcellular structures in the cytoskeleton and nuclei of fixed endothelial cells using femtosecond, near-infrared laser pulses focused through a high-numerical aperture objective lens (1.4 NA). We find that the energy threshold for photobleaching lies between 0.9 and 1.7 nJ. By comparing the changes in fluorescence with the actual material loss determined by electron microscopy, we find that the threshold for true material ablation is about 20\% higher than the photobleaching threshold. This information makes it possible to use the fluorescence to determine the onset of true material ablation without resorting to electron microscopy. We confirm the precision of this technique by severing a single microtubule without disrupting the neighboring microtubules, less than 1 µm away.