In this paper, we present a coupled experimental/theoretical investigation of pressure effect on the ferromagnetism of ${\mathrm{LaCrGe}}_{3}$ and ${\mathrm{LaCrSb}}_{3}$ compounds. The magnetic, electronic, elastic, and mechanical properties of ${\mathrm{LaCrGe}}_{3}$ and ${\mathrm{LaCrSb}}_{3}$ at ambient condition are studied by first-principles density-functional theory calculations. The pressure dependences of the magnetic properties of ${\mathrm{LaCrGe}}_{3}$ and ${\mathrm{LaCrSb}}_{3}$ are also investigated. The ferromagnetism in ${\mathrm{LaCrGe}}_{3}$ is rather fragile, with a ferro- to paramagnetic transition at a relatively small pressure (around 7 GPa from our calculations, and 2 GPa in experiments). The key parameter controlling the magnetic properties of ${\mathrm{LaCrGe}}_{3}$ is found to be the proximity of the peak of Cr density of states to the Fermi level, a proximity that is strongly correlated with the distance between Cr atoms along the $c$ axis, suggesting that there would be a simple way to suppress magnetism in systems with one-dimensional arrangement of magnetic atoms. By contrast, the ferromagnetism in ${\mathrm{LaCrSb}}_{3}$ is not fragile. Our theoretical results are consistent with our experimental results and demonstrate the feasibility of using first-principles calculations to aid experimental explorations in screening for materials with fragile magnetism.