Hi, Maia --
I promise that it allows you to be off more than a millisecond -- but you're right, it is relatively strict. The main reason for this is that a 16th note goes by pretty quickly, and your response needs to be divided up into 16th note chunks.
You're luck that I'm not supporting 32nd notes, or it'd be even harder!
The main thing is just practice (not just on the drill -- also just beating rhythms by yourself... try setting a metronome to just flash the light, and tap along with it... then close your eyes for 10 seconds or so. When you open them, are you still on the beat? It's harder than it sounds! You can have a friend test you as well, so you'll learn if you tend to speed up or slow down.
There's one trick that can help out -- the drills let's you be a little bit off the beat, as long as you're consistent. In other words, if you notice that your first note was a little behind the metronome click, then you can still get a correct answer if *all* of your notes are a little behind the metronome beat.
Rob