I think there are a couple of problems here. Each time you call your static methods that in turn creates a brand new instance of FileHandler class.
Instead
you should create only one instance of the FileHandler class for each log file and work with that.
As a side note, both of you static methods (getClientLogger and getServerLogger) are nearly identical. Sun may take points off for that since this is generally a bad programming practice.
Instead try creating only one method (getLogger) and make it take some kind of parameter that will differentiate between the different logging modes (client & server). Doing so will half the size of you FileLogger class.