An effort of proponents of relativity theory to find evidence for the so-called gravitational red-shift of spectral lines as one of the experimental consequences of Einstein's generalized theory of relativity is reconsidered with reference to hitherto unpublished documents. It is shown how much interest Albert Einstein in fact took, around 1920, in the data analysis of Leonhard Grebe and Albert Bachem, who tried to explain why most earlier efforts to find the gravitational red-shift had failed. They carefully measured the line (...) profiles of the spectral lines both in the sun's Fraunhofer lines and in pressure-independent laboratory comparison spectra. Then they rejected all lines with strong neighbouring lines which could cause apparent shifts of the line centres; the remaining seven well-isolated lines showed the gravitational red-shift with acceptable accuracy. Nevertheless, their claim to have successfully isolated the relativistic effect never convinced the astrophysicists of their day—the reasons for the scientific community's scepticism, contrasted against the enthusiastic group of Einstein sympathizers, are also discussed. (shrink)