GPS Error: Vertical vs. Horizontal

When we look at a GPS error value in a so-equipped GPS, does that value
reflect *both* the vertical (i.e. altitude) and the horizontal
component? Can a separate value for each be determined?

The reason I ask is that in firing up my Garmin 496 (an early Christmas
present, comments to follow in another post ), I noticed the altitude
initially register at 100 ft below actual then right itself after a few
minutes. All of this happened while the horizontal location did not
noticeably move (had it fully-zoom by the way) which led me to believe
that the error can be different from each other at least initially.

Marco