The paper is organized as follows. In Section 2 we briefly review linear response theory and the fluctuation-dissipation theorem. In Section 3 we propose a goodness-of-fit test to probe for the validity of linear response in time series. In Section 4 we discuss the logistic map, demonstrate the mechanism leading to the breakdown of linear response for this one-dimensional map and show how this breakdown might not be apparent with time series of insufficient length. We show the effect of finite **data** size as well as how the choice of the observable can either mask or emphasize the non-smoothness of the invariant measure. In Section 6 we show further that an application of the FDT in situations where linear response does not exist cannot provide any reliable statistical information, not even in an averaged sense. We conclude with a summary in Section 7.