Sometimes if the pickups are set too close to the strings, that can affect string intonation as well, but the magnetic fields of the pickup's pole pieces will exert a pull on the strings and make them read a bit flat, one other thing that can affect string intonation is the neck alignment with respect to the centre-line of the body, also, the straightness of the neck, if the neck is a bit twisted then the string intonation will be out in some parts of the neck too.
But definitely, if the string saddles are in the wrong place to begin with then the individual string intonation is going to be difficult to get it set correctly, I had first-hand experience of that when I tried to install a Floyd Rose Trem on my Gold Strat, I mistakenly positioned the bridge so that the saddles were positioned about 5mm further from the 12th fret than they should have been, this meant that the length of the string from the 12th fret was too long and consequently the 12th fret intonation of the strings was too flat, and all the string saddles had already been moved as far forward towards the 12th fret as they could go, but each string still read flat by at least 50 cents ( from memory).
Last edited by DrNomis_44; 20-02-2017 at 03:26 PM.