We prove better lower bounds on additive spanners and emulators, which are lossy compression schemes for undirected graphs, as well as lower bounds on shortcut sets, which reduce the diameter of directed graphs. We show that any $O(n)$-size shortcut set cannot bring the diameter below $Ω(n^{1/6})$, and that any $O(m)$-size shortcut set cannot bring it below $Ω(n^{1/11})$. These improve Hesse's [Hesse03] lower bound of $Ω(n^{1/17})$. By combining these constructions with Abboud and Bodwin's [AbboudB17] edge-splitting technique, we get additive stretch lower bounds of $+Ω(n^{1/11})$ for $O(n)$-size spanners and $+Ω(n^{1/18})$ for $O(n)$-size emulators. These improve Abboud and Bodwin's $+Ω(n^{1/22})$ lower bounds.