I have the Curve as well and it doesn't continue text; it is serviced by Sprint. One of my friends has a Storm on Verizon and they are not able to do that either. However my friend's Pearl on T-Mobile can text over the 160 character limit. I believe it has to do with the carrier.
As stated previously, the carrier does control this, but it's strange that your friend's VZW Curve is able to do this. Do you know if they've somehow changed any hidden settings or anything?