Fluorescence recovery after photobleaching (FRAP) is widely used to interrogate diffusion and binding of proteins in live cells. Herein, we apply two-photon excited FRAP with a diffraction limited bleaching and observation volume to study anomalous diffusion of unconjugated green fluorescence protein (GFP) in vitro and in cells. Experiments performed on dilute solutions of GFP reveal that reversible fluorophore bleaching can be mistakenly interpreted as anomalous diffusion. We derive a reaction-diffusion FRAP model that includes reversible photobleaching, and demonstrate that it properly accounts for these photophysics. We then apply this model to investigate the diffusion of GFP in He La cells and polytene cells of Drosophila larval salivary glands. GFP exhibits anomalous diffusion in the cytoplasm of both cell types and in HeLa nuclei. Polytene nuclei contain optically resolvable chromosomes, permitting FRAP experiments that focus separately on chromosomal or interchrosomal regions. We find that GFP exhibits anomalous diffusion in chromosomal regions but diffuses normally in regions devoid of chromatin. This observation indicates that obstructed transport through chromatin and not crowding by macromolecules is a source of anomalous diffusion in polytene nuclei. This behavior is likely true in other cells, so it will be important to account for this type of transport physics and for reversible photobleaching to properly interpret future FRAP experiments on DNA-binding proteins.