The early evolution of an outbreak of an infectious disease epidemic depends on its reproductive number R: the average number of secondary cases that a case generates. An accurate and timely estimate of the reproductive number is crucial to make projections on the near-future evolution of the epidemic, and to set up the appropriate public health response. Estimates of R often come from surveillance data, as it has been in the case of the COVID-19 pandemic. This means statistically inferring R from time series of daily reported cases, hospitalizations, or deaths. In this study, however, we argue that surveillance-based measures of the reproductive number may not always be accurate measures of the true reproductive number.