Windows 10 users have been reporting a lot of errors that say "requires at least 256 color mode". I've seen a lot of gobblety-gook solutions that seem to only apply to installing from disk and only apply to certain video cards, and only apply to windows 7 or 8.
The actual solution for me (windows 10, NVIDIA card) was to select "run as Administrator" and "Reduced Color Mode 8 bit 256" in compatibility settings. However this came with complications.
Symptoms
First, the game runs normal for me on first install. After a few times of opening and closing the application I get this error. Reinstalling will fix the error for a few times running the game. Fortunately you can keep your saves when you uninstall. However, this is a crappy solution.
The error is a blank screen followed by the "requires at least 256 color mode" dialog box. My desktop resolution gets set to a really low value and stays until I run a different game. Very odd indeed.
Complication
When trying to run the automatic compatibility troubleshooter, the compatibility settings are incorrectly set to 16bit 65,536 color mode and can never be changed. The reduced color mode drop down is grayed out an uneditable.
Work Around
Make a copy of the JK.EXE application in the same directory. Rename it something different (i.e. jk_copy.exe). Right click, select Compatibility tab, and set the reduced color mode to 8bit 256 color mode, and select run as Administrator.
Replace all of your Jedi Knight shortcuts or change them to point at this new EXE
Many tutorials have recommended adding -windowgui to the shortcut target. This didn't seem to do anything for me, but others might have more luck.