and just so we are on the same page, here are the trigger and test class i am using in a developer edition.  make sure your code is identical to mine for both the trigger and test class.  just put them in a sandbox, and click the run test button on the test class.

note i made a slight modification to the test class, though i don't think that's the issue.

i just ran the test on this test class again get 100% with no failures, so i have no idea how you are getting that error.  alternatively, you can just sign up for your own free developer sf edition, and put these in there, and you'll see what i'm talking about.  do you have any existing code on the contact object in your testing environment?

trigger:

trigger setTitle on Contact (before update) 
{    
    for(Contact c: trigger.new)    
    
    if(c.Title_Short_List_Job_Board__c != null && c.Title_Short_List_Job_Board__c != trigger.oldMap.get(c.id).Title__c)       
    {        
        c.Title__c = c.Title_Short_List_Job_Board__c;    
    }
}

test class:

@isTest
    private class testContactTrigger{
    static testMethod void verifyTitle(){

    Account a = new Account();
    a.Name = 'test1234';
    insert a;
    
    Contact c = new Contact();
    c.LastName = 'Test';
    c.AccountId= a.id;
    c.Title__c = null;
    c.Title_Short_List_Job_Board__c = null;
    insert c;
    
    c.Title_Short_List_Job_Board__c = 'title';
    update c;
    
    System.assertequals('title',c.Title_Short_List_Job_Board__c);        
    }
}