Another quirk is that pushing a string down onto a fret will change the tuning slightly. Plus, your strings can be too short or too long at the bridge, which will make them play out of tune at different spots on the neck. Having a nut that is too high can force you to push the string too far down to the frets too, which can make it out of tune.
Sometimes I have to hit a harmonic on a bass string so I can tune with my stage tuner, especially on low strings. It is a good idea to make sure your instrument has been carefully set up at the bridge with a good strobe tuner for this reason (and intonation reasons). I use a vintage Conn Strobo tube powered tuner.
