Consider a second order degenerate elliptic system whose principal part is given by the \(p\)-Laplacian \(\partial_ \alpha (| Du |^{p-2} \partial_ \alpha u)\). Problems of this type are usually studied in the Sobolev-space \(H^{1,p} (\Omega,\mathbb{R}^ N)\). The author shows that it is sufficient to assume \(u \in H^{1,p - \delta} (\Omega,\mathbb{R}^ N)\) for some \(\delta>0\) depending on the data in order to deduce \(u \in H^{1,p + \delta} (\Omega,\mathbb{R}^ N)\) which means that \(u\) is a classical weak solution. A similar result is obtained for systems of higher order. The proof makes use of Whitney extension theorem and Gehring’s reverse Hölder inequality.