A study into the maximum acceleration gradient, ${E}_{z}^{mathrm{max}}$, in a collinear dielectric wakefield accelerator due to single bunch beam breakup (SBBU) is presented. The longitudinal wakefield is proportional to charge over radius squared (${E}_{z}ensuremath{sim}Q/{a}^{2}$) which implies small $a$ is favorable for fixed $Q$. However, when the transverse wakefield (${E}_{ensuremath{perp}}ensuremath{sim}Q/{a}^{3}$) is also considered then the SBBU instability severely limits the charge that can pass through a long structure as required for a large transfer of energy from the drive bunch to the witness bunch and hence limits the sustained acceleration that can be obtained.
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