Recently, much progress has been made on techniques to measure the masses of new particles with partially-invisible decays at a hadron collider. We examine for the first time the realistic application of $M_{T2}$-based measurement methods to a fully hadronic final state from a symmetric two-step decay chain with maximal combinatorial uncertainty. Several problems arise in such an analysis: the $M_{T2}$ variables are powerful but fragile, with shallow edges that are easily washed out or faked by ubiquitous combinatorics background. Traditional methods of both cleaning up the distribution and determining edge position can fail badly. To perform successful mass measurements we introduce several new techniques: the edge-to-bump method of extracting an edge from a distribution by analyzing a distribution of fits rather than a single fit; a very simple yet high-yield method for determining decay chain assignments event-by-event; and a systematic procedure to obtain $M_{T2}$ edge measurements in the presence of heavy combinatorics background, the key element being the parallel use of at least two independent methods of reducing combinatorics background to avoid fake measurements. All of these techniques are developed in a Monte Carlo study of the decay $\tilde{g}\tilde{g}\to 2\tilde{b}+2b\to 4b+2{\chi }_1^0$ and verified in a second blind study with a different spectrum. In both cases, the gluino and sbottom masses are measured to a precision of $\sim 10\%$ with $\mathcal{O}(100{\mathrm{fb}}^{-1})$ at the LHC14 (assuming pessimistic $b$-tag efficiencies).

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