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Single-photon resolution (SPR) detectors can tell the difference between incoming wave packets of n and n+1 photons. Such devices are especially important for linear optical quantum computing with projective measurements. However, in this paper, I show that it is impossible to construct a photodetector with single-photon resolution when we are restricted to single-photon sources, linear optical elements, and projective measurements with standard (nonphoton-number discriminating) photodetectors. These devices include SPR detectors that sometimes fail to distinguish one- and two-photon inputs, but at the same time indicate this failure. PACS numbers: 42.79.Ta, 03.67.Hk, 42.79.Gn.