Recent studies have highlighted the ability of HIV to escape from cytotoxic T lymphocyte (CTL) responses that concurrently target multiple viral epitopes. Yet, the viral dynamics involved in such escape are incompletely understood. Previous analyses have made several strong assumptions regarding HIV escape from CTL responses such as independent or non-concurrent escape from individual CTL responses. Using experimental data from evolution of HIV half genomes in four patients we observe concurrent viral escape from multiple CTL responses during early infection (first 100 days of infection), providing confirmation of a recent result found in a study of one HIV-infected patient. We show that current methods of estimating CTL escape rates, based on the assumption of independent escapes, are biased and perform poorly when CTL escape proceeds concurrently at multiple epitopes. We propose a new method for analyzing longitudinal sequence data to estimate the rate of CTL escape across multiple epitopes, this method involves few parameters and performs well in simulation studies. By applying our novel method to experimental data, we find that concurrent multiple escapes occur at rates between 0.03 and 0.4 day(-1), a relatively broad range that reflects uncertainty due to sparse sampling and wide ranges of parameter values. However, we show that concurrent escape at rates 0.1-0.2 day(-1) across multiple epitopes is consistent with our patient datasets.