Anything big (like an airliner) isn't allowed to rely on radio altimeter alone for terrain avoidance for many years now. They're GPS augmented. Whenever it is confident where it is (which will be always or almost always) the plane is deciding whether it could hit something by continuing to move apparently forward and alerting the pilots about that.

When you hear "Obstacle Obstacle" announced in Sully, that's not a radio altimeter detecting the bridge, the computer has concluded that the plane's motion is uncomfortably close to the bridge tower based on GPS and a database of tall things you might fly a plane into (and thus it knows this is an "Obstacle" whereas a mountain is "Terrain").

In that movie there's not a damn thing they can do beyond avoiding the bridge itself, because they don't have any thrust - but ordinarily pilots should near instinctively try to ascend immediately for any of this group of warnings. The training is specific that this is not the time for diagnostics, if you aren't already absolutely sure why the computer thinks you're about to hit something you must assume the computer is correct and climb at best rate. This routinely saves lives when pilots become confused about where they are.