Why don't hyperbolic navigation transmitters on the surface (e.g. Loran-C) allow altitude measurement? I have been reading the following article about radio navigation in general and GPS / GLONASS receiver design:


A homemade receiver for GPS &amp; GLONASS satellites


in which the author states the following about hyperbolic radionavigation:


  Hyperbolic navigation systems were first implemented as ground-based navigation systems [...] Since the transmitters are located on the Earth's surface, the geometry of the problem does not allow a three-dimensional navigation. [...] To measure the altitude, one of the transmitters should be located above or below the user's receiver or at least out of the user's horizon plane. 


I do not seem to be able to comprehend why this is the case. Even if the transmitters are located on the surface (e.g. Loran-C), the differences between reception times of signals from 4 transmitters would generate 3 rotational hyperboloids in 3D space, the intersection of which could be above (or below) surface.
You are correct, the time difference generates hyperboloids in 3D space and as such can be used to find the 3D position. However, if the transmitters are far apart and/or in the same plane, the geometry of the intersecting hyperboloids is such that the accuracy in the vertical axis is very poor.

When the receiver is in the same horizontal plane as the transmitters, all intersections almost form a vertical line. The slightest inaccuracy in time difference, would cause only a slight error in the horizontal position, but a significant error in the vertical position. 

The concept is called Geometric Dilution of Precision (GDOP). A good start for more detail can be found on Wikipedia