# Problems from having too many interactions in a regression?

Excluding the 'dummy variable trap', are the problems from including too many interaction terms in a regression any different from the problems of including too many continuous or binary variables in a regression?

Furthermore, are the responses to this question conditional on interaction terms being continuous-with-continuous or binary-with-continuous?

If they involve categorical main effects they can lead to unexpectedly high loss of degrees of freedom as people often fail to take into account that the product of the degrees of freedom for the main effects is involved so if you have an interaction between a categorical variable with 5 levels and one with 6 you lose $$(5 - 1) \times (6 -1) = 20$$ more df.