I have no idea why you would like that, but [this](http://stats.stackexchange.com/questions/12484/constrained-linear-regression-through-a-specified-point) should contain what you are looking for. 

Are there a lot of fitted values $< 0$? If not then I do not think that you model is wrong (per se), it's just that the estimator (OLS?) can't fit the data (around 0) well. Most likely because there is not a lot of test scores at 0?

You could do exponential regression if indeed $Y$ is count variable; the tobit model is another way to go if you are looking for corner solution. But both of these models are harder (than OLS) to interpret - because they are non-linear - OLS is often the easy choice.

EDIT: Please note that when you force the intercept, there is no agreed upon way of calculating $R^2$ - and some might say that you cannot know what it's actually measure of.