Your action is too low or your neck is bowed out from too much truss rod tension, or you have a high fret.. Or a combination of all of the above.  Can you take some detailed pictures and post them up? Either way eye the neck and check if its straight, something like that should be obvious, then check your frets.
Cornford Hellcat
Peavey 5150
1994 Ibanez Jem 7V

My guitar teacher said it was the action. When he made it higher for me, it worked for about a week and then it just went back to what it does in the link. Then he fixed it again, and it still went back to that. He thinks it's a problem with my bridge.
It's really impossible to tell without having a good look at the guitar and seeing what is going on here. What guitar is it? What kinda bridge are we talking about.
Cornford Hellcat
Peavey 5150
1994 Ibanez Jem 7V
It sounds like a combination of your bridge simply being set far too low and a high fret.

I can tell there is a high fret because the frets sound totally dead for a couple of notes and then they return relatively to normal in the next fret.
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I also have to do that. Cottaging this weekend