saveStrings() just overwrite the current file with the new one, with the data given to the function.
Two possible fixes:
- Collect all data before calling saveStrings() (or save each time with the data cumulated so far);
- Use the Java API allowing to append to a file. I show an example at http://bazaar.launchpad.net/~philho/+junk/Processing/view/head:/_SmallPrograms/Email/Logger.java

1.    if(key=='r' && State == 'R'){
2.      println("R has been pressed");
3.    // Schrijf weg in textbestand dat Rood is ingedrukt bij stimulus i
4.       data[0] = "rood";
5.       saveStrings("data.txt", data);
6.       i = i + 1;
7.      // Als de 32 items bereikt zijn stoppen
8.      if(i>=Woorden.length) {
9.        
10.        //Laad Eindpagina en state naar D (Done) 
11.        State = 'D';
12.      }
13.   }

In this section, if (R)unning and 'r' pressed data[0] is set to "rood" , so your data[] array might look like this:

data[0]="rood"
data[1]=""
data[2]=""
etc

Next time this section is entered, exactly the same thing happens so your array still looks like this...
data[0]="rood"
data[1]=""
data[2]=""
etc


I think that's where the problem is - it shouldn't be too hard to fix, if you need help just shout!

Looking closer, indeed you save each string at its specific slot in the data array. But you declared this array within the keyPressed() function, so it is lost each time the function ends, and recreated each time it is called.
You should declare it outside of the function, at the global level.

anyone? 

Hello!

A longer explanation:

When you declare a variable, like -> String[] data = new String[32]; for example, that variable can only exist & be accessed within that function's scope.
In your case, the variable data was declared inside function keyPressed().
It means it can't be accessed by other functions, like setup() and draw() for example!
Plus, data is destroyed at the end of keyPressed()!!!
So the solution lies in declaring variables, which you need to be permanently accessed by everyone, in a global scope, outside of any function!

" If I put it outside the keypressed function, then how does the program know that pressing R should save red to the file."
It is unrelated.
All you have to do is to declare the array outside of the function. The function is still there, and can still use the same array, the only difference is that the values will persist between each key press.