Tuning issues - nut problem?

[Minininjarob]

Ok so I was playing yesterday my Player P Bass and for probably the first time I had the tuner on when I fretted an F. On the open E the note was perfectly in tune but the F was WAY off no matter how I fretted it - it was way too sharp. 
Thinking back I have always thought the first fretted notes especially the F sounded a bit weird but I not very experienced so just brushed it off. 
Is the nut too high? Trying to figure out what’s wrong and that’s the only thing I can think of. Rest of the intonation seemed relatively ok. 

[Doctor J]

A badly cut nut is the obvious culprit. How much of a gap is there between the bottom of the string and the top of the first fret? Are the other strings the same height off the first fret and showing the same problem? Have you tried a different string?

[Supernaut]

Check the intonation before looking at the nut. Try replacing the string too. 

[Minininjarob]

1 hour ago, Supernaut said:

Check the intonation before looking at the nut. Try replacing the string too. 

As I said in original post intonation already checked thoroughly. 

[Twigman]

You say intonation is good, at which fret?

How far up the neck do you have to go before it's in tune?

[lemmywinks]

If intonation is ok but it's sharp as early as the 1st fret then the nut is cut too high.