Deep convolutional neural networks (CNN) have previously been shown to be useful tools for signal decoding and analysis in a variety of complex domains, such as image processing and speech recognition. By learning from large amounts of data, the representations encoded by these deep networks are often invariant to moderate changes in the underlying feature spaces. Recently, we proposed a CNN architecture that could be applied to electroencephalogram (EEG) decoding and analysis. In this article, we train our CNN model using data from prior experiments in order to later decode the P300 evoked response from an unseen, hold-out experiment. We analyze the CNN output as a function of the underlying variability in the P300 response and demonstrate that the CNN output is sensitive to the experiment-induced changes in the neural response. We then assess the utility of our approach as a means of improving the overall signal-to-noise ratio in the EEG record. Finally, we show an example of how CNN-based decoding can be applied to the analysis of complex data.