A few times when someone close to me had died (but I didn't know about it yet), or was about to die, I got pretty damn agitated/restless... Much more so then usual.
Other times, I pretty much "knew" that it had happened. Of course, these were for older relatives, and it was expected any day, I just happened to call it.
'Course, these things have only happened a half dozen times or so, so I wouldn't call it a 6th sense yet. I'll get back to you when I'm 50 or so and see if I've changed my mind on that.
However, I wouldn't doubt the possibility - there have been a number of documented incidents of animals (and people for that matter) clearly knowing "something was up" when they really shouldn't have been able to.