Prediction interval behaviour close to the origin

I have a simple linear regression model, for which I have calculated the Confidence and Prediction Intervals.
The model is fixed at the origin (regression line goes through 0,0), and so the prediction interval should "pinch" together and go through the origin as well according to my understanding. After all, new "0" observations of the predictor variable should not imply a possibility of a range of values for the response should be 0 as well.
The same can be said of negative predictions...they should not be allowed.
But the prediction interval I have is fixed for the whole sample range, and dips below the x-axis. I can't seem to find any good explanations of the PI, so any hints or suggestions as to where to look would be greatly appreciated.