Lithium in the atmosphere of a F or G supergiant reflects the initial Li abundance and the internal history of the star. During evolution of a star from the main sequence (MS) to the supergiant phase, lithium may be destroyed by, for example, rotationally induced mixing in the MS stars and strongly diluted by development of the supergiant's convective envelope. In order to probe the connection between atmospheric Li abundance and evolutionary predictions, we present a non-local thermodynamic equilibrium abundance analysis of the resonance doublet Li i at 6707.8 Å for 55 Galactic F and G supergiants and bright giants (we observed 43 of them, the remaining 12 are added from Luck and Wepfer's list). The derived lithium abundances log ε(Li) may be considered in three groups, namely: (i) 10 Li-rich giants with log ε(Li) = 2.0–3.2 (all 10 are F-type or A9 stars); (ii) 13 G- to K0-type stars with Li abundances in the narrow range log ε(Li) = 1.1–1.8; (iii) all other stars provide just upper limits to the Li abundance. The derived Li abundances are compared with theoretical predictions of 2–15 M⊙ stars (both rotating and non-rotating). Our results are generally in good agreement with theory. In particular, the absence of detectable lithium for the majority of programme stars is explainable. The comparison suggests that the stars may be separated by mass M into two groups, namely M ≲ 6 M⊙ and M > 6 M⊙. All Li-rich giants and supergiants with log ε(Li) ≥ 2.0 have masses M < 6 M⊙; this conclusion follows not only from our work but also from a scrutiny of published data. 11 of 13 stars with log ε(Li) = 1.1–1.8, specifically the stars with M < 6 M⊙, show good agreement with the post-first dredge-up surface abundance log ε(Li) ≈ 1.4 predicted for the non-rotating 2–6 M⊙ stellar models. An absence of Li-rich stars in the range M > 6 M⊙ agrees with the theoretical prediction that F and G supergiants and giants with M > 6 M⊙ cannot show detectable lithium. We note that present theory appears unable to account for the derived Li abundances for some stars, namely for (i) a few relatively low-mass Li-rich giants (M < 6 M⊙), whose high Li abundances accompanied by rather high rotational velocities or substantial nitrogen excess contradict theoretical predictions; (ii) the relatively high-mass supergiants HR 461 and HR 8313 (M > 6 M⊙) with the detected abundances log ε = 1.3–1.5. It is possible that the lithium in such stars was synthesized recently.