It is claimed that the accuracy of the computation of the eigenvectors of a real symmetric tridiagonal matrix by the EISPACK routine TINVIT can be improved by computing each eigenvector from a different random starting vector and by performing an additional iteration after the stopping criterion is satisfied, leading to result competitive in quality with divide-and-conquer and QL methods. The claim is supported by a statistical analysis and numerical experiments with matrices of order up to 525, with some discussion for larger-order matrices.