In a multiple regression analysis what would likely be the culprit of a dramatic increase in the p-value of one of the variables?

This can happen if you add another independent variable to your regression model that is strongly correlated to some other variable already in the model.
This is called multicollinearity. 
If there is a high correlation between your independent variables can lead to problems.
It can lead to increased variance of the coefficient estimates and make the estimates very sensitive to minor changes in the model.

A. is the answer in this question