Addition of independent variables will cost degrees of freedom but will also have added explanatory power and thus, increasing the ESS and decreasing the RSS.

increase in ESS leads to increase in EMS and decrease in RSS will lead to decrease in RMS as well. with the increase in the number of independent variables, there will be an increase in degree of freedom for RMS  and a decrease in degree of freedom for EMS. so this will further decrease F-stat...

i don't know why you say it could be increasing....

or am i missing sth? or are you just throwing the darts in the dark and trying to be safe stating it could both increase or decrease? hahahaha