We consider the properties of the bootstrap as a tool for inference concerning the eigenvalues of a sample covariance matrix computed from an $n\times p$ data matrix $X$. We focus on the modern framework where $p/n$ is not close to 0 but remains bounded as $n$ and $p$ tend to infinity. Through a mix of numerical and theoretical considerations, we show that the bootstrap is not in general a reliable inferential tool in the setting we consider. However, in the case where the population covariance matrix is well-approximated by a finite rank matrix, the bootstrap performs as it does in finite dimension.