It could be that the original cushions have fallen off. You can usually find replacements at any general store but if you prefer not to make the trip and you happen to have Sugru around, replacing a cushion is easy!
Just roll up a small ball of Sugru about the size of a large pea and place it in the same location as your missing cushion. I lightly closed the door just to flatten out the surface of the Sugru but whatever you do, don't leave the door closed. Your Sugru could end up sticking to both sides and gluing your cabinet shut.
If you did everything correctly, the next morning you should have a nice, quiet cabinet door.