I actually saw "The Understudy" the other day, with a SM friend of mine. Julie White plays the stage manager, and is quite good in the role! There is a great deal of the stage manager being emotional, etc. throughout the performance - however it is written into the script.
Basically it's a 3-person show, the other two actors are Mark Paul Gosselaar & Justin Kirk (I can't remember the characters names and don't currently have the program handy). The show takes place during an understudy rehearsal, where Justin Kirk plays the Understudy. There is a past relationship between the understudy & the stage manager and a few other reasons why she is thrown for a loop with him being there thus catching her off-guard and more emotional than one would expect of a stage manager. There are many times throughout the play where the actor (Mark Paul Gosselaar) does point out that the stage manager usually doesn't act like this, and details the reasons why she is so great at her job. While she is emotional, she does manage to continue to fulfill her duties as the stage manager.
Overall, my friend & I both really enjoyed the show and there was enough "insider" stuff to make us happy, but I think a lot of it also carries through to people that are not in the industry.