Yes, it is possible but not as straightforward as the question may indicate. Points are only one of many items needed. For example, you have the points but do you have the names, start and end point of a road distinguished, z-level information (one road may cross another but not have an intersection), one-way information, and so on? If you have points then you have points that exist most likely on streets but also in driveways, parking garages, etc. What I am saying is that you have points but none of the other vast amounts of information needed to make streets. Some start points:
1) Use an already existing road product this is freely available and “adjust” the geometry of this product to your points. Even now, you really need to convert your points to centerlines so you need the number of lanes, divided highway status, and so forth.
2) You can “connect” the points and make lines based on their time-stamp in sequence but even then, you need to know the start and end of each street to break them. You can do this is QGIS using http://spatialgalaxy.net/2011/12/30/qgis-plugin-of-the-week-points-to-paths/
3) The driver of the vehicle needs to know they were going to be converted to streets so they drove consistently in one lane, did not stop at roadside stores, etc., or swing U-turns etc.
You can turn GPS points into lines easily enough using (2) but to convert them to streets or roads