Debugging isn't a reliable method for checking things in a unit test. Use assertions:
Account reloaded = [SELECT Id, managed_package_custom_field__c FROM Account WHERE Id = :acct.Id ]);
Presuming this fails, there may be a trigger on Account by the managed package, or maybe a workflow rule cancelled out the value, there are some missing Custom Settings, or something else happened entirely. You'll have to read more than just a single line of log to figure out what happened.
If you open the debug log in Developer Console, you can type in the custom field's name in the Filter box, and from there, you can see each time it was touched and what happened to it. If you don't find anything there, you might also need to work with the ISV so they can get ISV-enabled log files, or, you might just ask them; they might already know of this problem and have a workaround for you.