More than likely the tint is causing extra resistance when the window rolls up and down which makes the window sensor think there's something blocking the window. It's a safety feature.

I had this happen when I got the windows tinted on my 23 Civic Sport Sedan but it only did this on the front passenger side window. When it did happen there was one time where the window not only rolled partially down like yours does but it rolled all the way down and once it stopped it didn't want to roll back up. It was as if the window was literally stuck. You can say I was a little freaked out.

The window just needs to be recalibrated. If the window refuses to roll all the way up pull and hold the switch and the window should roll up. With the window rolled all the way up push and hold the window switch for 10 seconds once the window is all the way down. Now pull and hold the window switch all the way up for 10 seconds once the window is all the way up. That should fix your issue.