Relative dimensions of the isotypic components of the N-th order tensor representations of the symmetric group on n letters give a Plancherel-type measure, called the Schur-Weyl measure, on the set of Young diagrams with n cells and at most N rows. We obtain logarithmic, order-sharp bounds for the maximal dimensions of the isotypic components of the tensor representations and prove that the typical dimensions, after appropriate normalisation, converge to a constant with respect to the Schur-Weyl measures.