The theory of low-rank tensor-train approximation is well understood when the approximation error is measured in the Frobenius norm. The entrywise maximum norm is equally important but is significantly weaker for large tensors, making the estimates obtained via the Frobenius norm and norm equivalence pessimistic or even meaningless. In this article, we derive a direct estimate of the entrywise approximation error that is applicable in some of these cases. The estimate is given in terms of the higher-order generalization of the matrix factorization norm, and its proof is based on the tensor-structured Hanson--Wright inequality. The theoretical results are accompanied by numerical experiments carried out with the method of alternating projections.