What if my force vs. length graph for a spring is weird for small displacements?

Tim and Andy measuring the force applied by a spring

I think every physics class in the known universe does the F vs. x experiment for a spring:  The force on a spring is measured with a spring scale or hanging masses, and is plotted on the vertical axis of a graph.  The length of the spring (or the displacement from the resting position) is measured with a meterstick and plotted on the horizontal axis.  Because F = kx, the slope of this linear graph is the spring constant k.  

(As an aside, I've written up a detailed approach to this experiment for the College Board -- take a look here.)

This experiment is beautiful because the data are easy to take, and because even the worst experimenters get something resembling a line.  However, occasionally you'll see something weird -- the graph will be a line most of the way, but very small displacements will give a significantly steeper slope.  See the graph to the right (and click on it to enlarge if you can't quite see).  

What's going on?

First of all, quash the inevitable misconception:  "Oh, that makes sense because the more the spring stretched, the more force we had to use."  Well, of course -- that's what F = kx means.  We should need more force to stretch the spring for larger displacements.  

The slope of this graph represents the spring constant k, which indicates the stiffness of the spring. What's happening here is that the spring is significantly stiffer under about 3 cm of stretch.  Does that make any physical sense, though?

Well, in this case, yes.  If you get this sort of data, take a careful look at the spring you're using:

See how many of the coils are touching each other?  I asked the class to be very quiet... and then I began to stretch the spring a couple of centimeters.  We could all hear the "poing!" sounds of the individual coils unsticking from each other.  All the coils were fully separated when I had stretched the spring... about 3 cm.