Marco Leon writes:

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?

No. The vertical error is usually much greater than the horizontal
error. The displayed error, if there is only one, will generally be
the horizontal error.

Can a separate value for each be determined?

Yes, but it's only practical for the computer to do it.

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.

Initially and continually. GPS altitudes should be treated as general
indications, not as precise altitudes that can be substituted for
altimeter readings.

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