The pseudospectrum is a tool to investigate the sensitivity of eigenvalues to perturbations of the corresponding matrices. The concept has been widely applied, however most of the applications of pseudospectra consider non-normal matrices, which exclude self-adjoint systems implemented using symmetric matrices. However, the most interesting case for pseudospectra is when there is a lack of an orthogonal basis of eigenvectors. Recent results have shown that most viscously damped second order linear systems with repeated eigenvalues are defective, that is they do not have a full set of independent eigenvectors. In practice eigenvalues will not be repeated, but will be clustered closely together in frequency. Although theoretically a full basis of eigenvectors exists, studying the pseudospectra of defective systems is vital to understand the potential computational problems for systems with clustered eigenvalues, when the eigenvector matrix may become ill-conditioned. This paper will investigate how the pseudospectra change as a system transforms from one with well separated eigenvalues, through to a defective system. This approach will yield insight into the problems associated with the computation of clustered eigenvalues.