Update: I may have fixed it.
I tried selecting the tabs in the layout mode, dragging the whole lot of them off the page and back on (I did this to see if there were any fields hidden behind it that might be somehow causing the problem - there weren't).
However, when I went back into browse, it suddenly displayed the way it was supposed to (ie when I am on the first tab, only the fields I have set out on that tab display). I'm not sure why this would fix the problem - sounds like a bug to me.
This happens when the top pixel of the tab object is in the header part.
Good to know, but not the problem in this particular case.
Thanks so much for posting! While this might not be the answer for the original poster, this was EXACTLY the problem I was having - turns out my tabs were indeed just a few pixels too high and strayed into the header. This has been driving me crazy for weeks, but now I know!