This is one of those problems that can get you pulling your hair!
There is whole list of things that could cause it and you can’t just point to one and say ‘this is where the problem lies’. So, the way to go (at least I did), is to start at the beginning and go through the steps, loosening and tightening strings after each step, as you go along.
Unfortunately, there is no conviction in what you do, only hope that the step you are at will be the one that will banish that awful buzz. Side note: If you think that this is the step you can jump to and solve the problem once the next instrument with such a problem comes along, you are mighty wrong, my friend.
So, after that rather rambling preamble, I think you got my drift: one such instrument came to me. The 13th fret was buzzing on the thin ‘e’ string.
Unconsciously, I ‘pooh-pooed’ it and tightened the truss rod, half a turn. Problem solved? Nope! Then, I gave it another half turn. Solved? Nope!
As the neck began developing a back-bow, I released the truss rod a bit and decided to shim the saddle under the area where the first string sat, thinking that raising it would raise the entire string a wee bit. Solved? Nope!
Then I decided to spot dress the 14th fret. Sanded, crowned and polished it. Solved? Nope! In fact, now the ‘e’ string had developed a buzz on the 14th and 15th fret too!!!!!
Then I decided that the whole fretboard – at least on the treble side – needed a fret-dress. So that was done. Solved? Nope!
Then I thought maybe it was a single fret that was popped up. I tapped down every, single fret on the treble side. Solved? Nope!
As I stood scratching my head, I recalled that I had neither dressed the zero fret, nor did I hammer it in, so I did that. Solved? Nope!
Then, I looked at the nut, squinting with suspicion at it. The grooves were indeed deeper than I would have liked to see. So I did the baking soda-super glue trick ( a pinch of baking soda in the string groove with a drop of superglue holding it in). Problem solved? Nope!
Then I decided to pull out the old nut and put in a new one. So, out it came and I sat a new one in its place. But the new one would not sit properly at the treble end.
I looked at it and sure enough, there was a remnant of the old nut stuck there, as if not wanting to go away. Pins and needles refused to make it budge and I had to pull out my freshly sharpened chisel to dull its edge on it. After I stuck the new nut in place did that solve the problem? Nope!
Then, as I ran out of options, I decided to give the sides of the frets (along the side of the fretboard) a shave. They were sticking out but nothing that would send you rushing into the ER. Problem solved? Nope!
Then as I stood exasperated, I took off the strings one more time and began dressing the 14th, 15th 16th and 17th fret with a vengeance: sanded, crowned and polished them shiny new and strung up the guitar again. Problem solved? Yes!
Yup! Problem solved. But hadn’t I done this before? I did, but maybe not enough the first time round.