A search for vector-like quarks and excited quarks in events containing a top quark and a $W$ boson in the final state is reported here. The search is based on $20.3 {fb}^{-1}$ of proton--proton collision data taken at the LHC at a centre-of-mass energy of 8 TeV recorded by the ATLAS detector. Events with one or two leptons, and one, two or three jets are selected with the additional requirement that at least one jet contains a $b$-quark. Single-lepton events are also required to contain at least one large-radius jet from the hadronic decay of a high-$p_\mathrm{T}$ $W$ boson or a top quark. No significant excess over the expected background is observed and upper limits on the cross-section times branching ratio for different vector-like quark and excited-quark model masses are derived. For the excited-quark production and decay to $Wt$ with unit couplings, quarks with masses below 1500 GeV are excluded and coupling-dependent limits are set.