You can check teh nut height. If the nut slots are cut shallow (i.e. the strings are sitting high at the nut) then intonation will be a bit sharp. You wont get a hell of a lot out of it though.
This is also assuming that the action is as low as is practicable. If its still quite high when its at its lowest then a fret dress can usually get rid of a lot of the buzzing and give you more wiggle room (even if the frets are unworn; new guitars often need fret dresses because, while the frets are probably damned near perfectly level when put in, when the necks strung up for the first time the stress on it forces some frets out a little bit, and that can go unaddressed for years).
Also, since the problem is high on the neck, try the neck relief - get it as flat as you can. I set relief so that the neck is as close to flat as practicable, and do it by getting it so when shining a light down the edge of the fretboard and looking down it it seems to be flat, and holding down the low E/whatever youre tuned to at 1st and 15th the gap between the E and 7th fret is barely visible, or even invisible, BUT that if I tap the string lightly when held down it makes a clack against the fret, so that I know there is actually a tiny bit of curvature (which is just so that I know that the neck isnt bowing backwards at all - like I said as close to flat as practicable).
So, lower action, fret dress if need be, deepen nut slots if possible to do so (carefull, you cant put material back in them!), level the neck and then see if youre getting better intonation.