I'm having trouble with my Epiphone SG G400 Pro. The 3rd string is sharp whenever I fret from the 1st - 12th frets. It's a tune-o-matic bridge and the saddle is as far from the nut as it can go. It's strung with the set of 10s that came on it from the factory. If it were the truss rod, wouldn't that affect all the strings? What might the problem be? The nut? Just been built badly?
It's the nut. Both the nut and the saddle affect Intonation. A compensated nut is used in higher end guitars to make adjustments for this. Moving the saddle with a tune-o-matic affects each string proportionally. By changing the saddle each note is shifted relative to it's starting position.
There are a variety of problems in how the guitar was built that would lead to sloppy intonation.
I agree with max dickinson: likely to be the nut.
If it's sharp all the way up the neck, when open tuning is fine, then the problem is probably the 'reference point' it's starting from (the nut).
It's possible that the groove in the nut for the 3rd string is angled such that the string is in fact resting on the far end of the nut, rather than the bridge end.
That is: The string comes from the tuning peg, acrodss the nut and rests on a high point. Usually that high point is nearest the frets, but if it's cut or worn badly then it could be that the high point is nearer the tuning peg, thus making the intonation ourt for the whole fretboard.
I'm not sure how you can check this other than a visual inspection