| Your nut slots should be not be so deep as to allow the open string to buzz, which is generally just a tiny bit higher than the first fret. Getting the slots filed to the correct depth is an iterative process - file, measure, file some more, measure some more, etc., til you get it right. Here's how I do it:
Before you've filed the slot all the way, with all strings on and the bass in tune, press down the subject string just in front of the second fret (that is, not where you'd ever want to fret a note, the opposite side of the second fret) so that the string makes a straight line between the top of the second fret and your nut slot. There should be a teeny tiny gap between the string and your first fret. If the string is touching the first fret, the slot for that string is likely too deep (but again, the real factor is if the open string buzzes, so if it doesn't, you're cool). If the string is still some distance above the first fret, file the slot some more, repeat until the string is nearly (but not quite) touching the first fret.
If you go too low, you either have to start over with a new nut (nah), fill in the slot and try again (rather not), or put a shim under the nut (better option, I had to put a single little layer of masking tape under the bass side of a nut once, hasn't been a problem). The best option is of course to not file it down too deep, stop before you take off too much material.
This is the method that's worked for me but I'm not an expert, so someone who is might correct me here.
And remember that the bottom of the nut slots shouldn't be parallel with the fretboard, but must angle back towards the headstock just slightly, and then the string should angle down even further to the post (or string tree, depending on which string); in other words, there should be a witness point on both the fretboard and headstock sides of the nut. | 
