Yes, its possible to download all of the GPS tracks for an area from OSM.
For a fairly small area, easiest way is probably to use the JOSM editor. You can just draw a box for the area you want, then download all of the tracks in that area. Then you can save that as a GPX file.
Or you can use the API in some other software, see the documentation here: http://wiki.openstreetmap.org/wiki/API_v0.6
Or you use the Planet.gpx, which has extracts for complete countries etc. Though this is just the coordinates, it doesn’t have any time data or tags etc. http://wiki.openstreetmap.org/wiki/Planet.gpx
Note the coverage/density of GPS tracks in OSM may vary. The main reason for uploading tracks is to assist with mapping. So if an area is already well mapped, people probably won’t bother to upload new GPS tracks. Also many people just use their own GPS tracks on their own computer to map for OSM, without actually uploading the tracks. So even in big cities, there might not be many tracks available.
Also consider the type of traffic. Many tracks will be from walking/cycling, and they may have a lot of stopping to survey things, or wandering off in random directions etc. So they may not represent typical traffic behaviour. Some tracks have tags, which can tell you what sort of vehicle it is.