Context. Chemical fractionation reactions in the interstellar medium can result in molecular isotopologue abundance ratios that differ by many orders of magnitude from the isotopic abundance ratios. Understanding variations in the molecular abundance ratios through astronomical observations provides a new tool to sensitively probe the underlying physical conditions. Aims. Recently, we have introduced detailed isotopic chemistry into the KOSMA-tau model for photon-dominated regions (PDRs), which allows calculating abundances of carbon isotopologues as a function of PDR parameters. Radiative transfer computations then allow to predict the observed [C II]/[C-13 II] line intensity ratio for specific geometries. Here, we compare these model predictions with new Herschel observations. Methods. We performed Herschel/HIFI observations of the [C II] 158 mu m line in a number of PDRs. In all sources, we observed at least two hyperfine components of the [C-13 II] transition, allowing determination of the [C II]/[C-13 II] intensity ratio, using revised intrinsic hyperfine ratios. Comparing the observed line ratios with the predictions from the updated KOSMA-tau model, we identify conditions under which the chemical fractionation effects are important, and not masked by the high optical depth of the main isotopic line. Results. An observable enhancement of the [C II]/[C-13 II] intensity ratio due to chemical fractionation depends mostly on the source geometry and velocity structure, and to a lesser extent on the gas density and radiation field strength. The enhancement is expected to be largest for PDR layers that are somewhat shielded from UV radiation, but not completely hidden behind a surface layer of optically thick [C II]. In our observations the [C II]/[C-13 II] integrated line intensity ratio is always dominated by the optical depth of the main isotopic line. However, an enhanced intensity ratio is found for particular velocity components in several sources: in the red-shifted material in the ultracompact H II region Mon R2, in the wings of the turbulent line profile in the Orion Bar, and possibly in the blue wing in NGC 7023. Mapping of the [C-13 II] lines in the Orion Bar gives a C+ column density map, which confirms the temperature stratification of the C+ layer, in agreement with the PDR models of this region. Conclusions. Carbon fractionation can be significant even in relatively warm PDRs, but a resulting enhanced [C II]/[C-13 II] intensity ratio is only observable for special configurations. In most cases, a reduced [C II]/[C-13 II] intensity ratio can be used instead to derive the [C II] optical depth, leading to reliable column density estimates that can be compared with PDR model predictions. The C+ column densities show that, for all sources, at the position of the [C II] peak emission, the dominant fraction of the gas-phase carbon is in the form of C+.