ROC sensitivity and specificity

In all the ROC curves I've seen the sensitivity increase as the (1-specificity) increases, is this always the case? I guess my question can be summarized as: can the fpr be higher than the tpr?

On the ROC plot, the boundary line $fpr = tpr$ corresponds to the case where the positive and negative distributions overlap completely.
The $tpr$ tracks higher than the $fpr$ when there is some separation between the two distributions.
And if the $fpr$ is ever tracking higher than the $tpr$, one would simply reverse the sense of the binary classification.