You can see from the following code that there is no point where the three circles intersect. If you want cities within those circles, though, the San Antonio requirement is redundant: as you can see, the intersection between the Houston and El Paso disks is the region of interest:
GeoGraphics[{ GeoDisk[Entity["City", {"Houston", "Texas", "UnitedStates"}], Quantity[333, "Miles"]], GeoDisk[Entity["City", {"SanAntonio", "Texas", "UnitedStates"}], Quantity[755, "Miles"]], GeoDisk[Entity["City", {"ElPaso", "Texas", "UnitedStates"}], Quantity[460, "Miles"]]} ] 
One would hope to calculate the intersection of those regions, then lookup cities in that region. Unfortunately, however, there is no GeoRegionIntersection function (see Find the GeoArea of overlapping GeoDisks and GeoArea of an Intersection).
A second approach would be to get the list of all cities in those two relevant regions, then take the intersection of those lists:
houston = GeoNearest[ "City", Entity["City", {"Houston", "Texas", "UnitedStates"}], {All, Quantity[333, "Miles"]} ] ... and a similar one for El Paso. Unfortunately again, though, that command times out. It does work for a smaller radius (e.g. 20 miles or so), so it's just a humongous list and my system / connection chokes on it.
If you are able to get the lists above, then Intersection[houston, elpaso] would do the trick.