Can't place MIDI-note exact on beat

[nothe]

When I zoom in realy much on in MIDI Editor on any of my MIDI or instrument tracks, I can see that all of my MIDI notes are a tiny bit of beat. And when I select Slip Mode and move the note, I can't put it on beat. I can only choose between a tiny bit before the beat or a little bigger tiny bit after the beat.
I've tried to quantize and use Grip Mode, but nothing helps.
When I change the track to use Samples instead of ticks, the notes move a tiny bit, but they dont move on beat. They just move randomly. I am talking about very very short times, not hearable. The problem is that this strange bahavior causes the regions to delete the first note sometime, because the first note i actualy about a quarter tick outside of the region.

Here is a screen shot of where I can slide the note:
http://tvattnoten.se/midiproblem.jpg

What do I do?
I know that there is a workaround for the problem. If I enlarges the region a bit, the first note doesnt disappear, but then I got other problems. So I need a solution, not a workaround.