I have a knock-off telecaster made by Stadium. The strings on it refuse to be in tune with themselves (the G and B strings especially). I hooked it up to a tuner and tested the intonation. The G string when open reads as G, at the 12th fret it also reads as G, The 12th fret harmonic reads G as well. When playing any other fret on that string however it is not an exact note. The notes are off by close to a quarter tone (50 cents) at some places, the 2nd and 3rd frets are particularly bad. I'm at a loss to what is wrong here.
If it's an exceedingly low-quality product, it could just be that the fretboard is badly designed enough that the notes are just not in the right place, but realistically, all guitars exhibit tuning issues with fretted notes. Equal temperament is a compromise to begin with, and the guitar itself even more so.
Assuming it's not actually a manufacturing defect, guitarists have over the years tried to mitigate the problem. If I'm not mistaken, Eddie Van Halen liked to tune his B string slightly flat so that the major chords he played sounded better in tune.
(Relatedly, this is one reason power chords are so prevalent in guitar music -- aside from being easier to play, they also use few enough notes that tuning issues don't arise.)
The only real answer to this whole issue has arisen relatively recently thanks to advances in fretboard craftsmanship. The True Temperament system provides you with a neck built specifically for your instrument that moves the fret location for each note so that it is best in tune. Steve Vai in particular has taken a liking to them.
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I don't know if this is the case here, but my first bass was an extremely cheap build, and I had something similar to this happening.
What I think may be happening is that you're fretting on your lower fret (Lets say 12 for sake of argument), and it may be playing the tone that should be heard at 13, or 14 or 15, etc... as if you were fretting that note. What causes that is often that there is a height difference between each fret, and the string is fretting itself on the higher of the two frets, this usually only affects a range of one to two frets before that fret, and on VERY rare occasions, 3 frets.
By (VERY CAREFULLY) filing the higher fret down slightly, you may be able to alleviate the problem without spending a ton of money on a (possibly) cheap guitar. If you're not comfortable doing this yourself, or you just plain don't know how to, find a instrument repair shop, and have them do it, you can try asking if you can sit in while the repair is done, but the answer will vary from shop to shop. Basically, it comes down to your skill level and budget.
Again, this may be completely off, but I had to make this modification myself when I first started, and your situation sounds similar to what I dealt with.