Hi Tobi and welcome,
If you are using a browser-based form, you may need to change your button properties' browser post back settings to "Always" to get it to work. We have seen issues with buttons not working because of that. InfoPath tries to intuit when it should post back to the server but sometimes it guesses wrong.
If it's not a browser form, it will be something to do with your rule. Add a dialog popup action after your submit action to verify the rule is firing.
Good luck!