It just depends how it's been generated. If it's a recorded track, it will usually have elevation data embedded. If it's a generated one (from a planner) then it usually won't. But neither are hard and fast rules, and Bikehike perhaps does add elevation to the files you download (including gpx), I don't know, don't use it.
In any case this is just a party trick really - generally you will be wanting to see something more useful on screen - for example in the Dakota screenshot I posted above, the elevation plot prevents the usual navigational messages from appearing, so I doubt if anyone actually uses it.