In two recent papers, Albrecht and White [‘Counting paths in a grid’, Austral. Math. Soc. Gaz.35 (2008), 43–48] and Hirschhorn [‘Comment on “Counting paths in a grid”’, Austral. Math. Soc. Gaz.36 (2009), 50–52] considered the problem of counting the total number Pm,n of certain restricted lattice paths in an m×n grid of cells, which appeared in the context of counting train paths through a rail network. Here we give a precise study of the asymptotic behaviour of these numbers for the square grid, extending the results of Hirschhorn, and furthermore provide an asymptotic equivalent of these numbers for a rectangular grid with a constant proportion α=m/n between the side lengths.