Multiple testing issues have long been considered almost exclusively in the context of General Linear Model, in which usually the significance of a quite limited number of contrasts is tested simultaneously. Most of the procedures used in this context have been designed to control the so-called Family-Wise Error Rate (FWER), defined as the probability of more than one erroneous rejection of a null hypothesis. In the last two decades, large-scale significance tests encountered for example in microarray data analysis have renewed the methodology on multiple testing by introducing novel definitions of Type-I error rates, such as the False Discovery Rate (FDR), to define less conservative procedures. High dimension has also highlighted the need for improvements, to guarantee the control of the error rates in various situations of dependent data. The present article gives motivations for a factor analysis modeling of the covariance between test statistics, both in the situation of simultaneous tests of a small set of contrasts in the General Linear Model and also in high-dimensional significance tests. Impact of the dependence on the power of multiple testing is first discussed and a new procedure controlling the FWER and based on factor-adjusted test statistics is presented as a solution to improve the Type-II error rate with respect to existing methods. Finally, the beneficial impact of the new method is shown on simulated datasets.