The King James translates the same verse as this:
But made himself of no reputation, and took upon him the form of a servant, and was made in the likeness of men:
Of course, there are likely to be several different meanings attributed to this, but the most common is the one that you posted - that He took on the humble nature of a servant.
Barnes' notes on the Bible says this: (Emphasis mine where added)
But made himself of no reputation - This translation by no means
conveys the sense of the original According to this it would seem that
he consented to be without distinction or honor among people; or that
he was willing to be despised or disregarded. The Greek is ἑαυτον
ἐκένωσεν heauton ekenōsen. The word κενόω kenoō means literally,
to empty, "to make empty, to make vain or void." It is rendered: "made
void" in Romans 4:14; "made of none effect," 1 Corinthians 1:17; "make
void," 1 Corinthians 9:15; "should be vain," 2 Corinthians 9:3. The
word does not occur elsewhere in the New Testament, except in the
passage before us. The essential idea is that of bringing to
emptiness, vanity, or nothingness; and, hence, it is applied to a case
where one lays aside his rank and dignity, and becomes in respect to
that as nothing; that is, he assumes a more humble rank and station.
In regard to its meaning here, we may remark:
(1) that it cannot mean that he literally divested himself of his
divine nature and perfections, for that was impossible. He could not
cease to be omnipotent, and omnipresent, and most holy, and true, and
(2) it is conceivable that he might have laid aside, for a time, the
symbols or the manifestation of his glory, or that the outward
expressions of his majesty in heaven might have been withdrawn. It is
conceivable for a divine being to intermit the exercise of his
almighty power, since it cannot be supposed that God is always
exerting his power to the utmost. And in like manner there might be
for a time a laying aside or intermitting of these manifestations or
symbols, which were expressive of the divine glory and perfections.
(3) this supposes no change in the divine nature, or in the essential
glory of the divine perfections. When the sun is obscured by a cloud,
or in an eclipse, there is no real change of its glory, nor are his
beams extinguished, nor is the sun himself in any measure changed. His
luster is only for a time obscured. So it might have been in regard to
the manifestation of the glory of the Son of God. Of course there is
much in regard to this which is obscure, but the language of the
apostle undoubtedly implies more than that he took an humble place, or
that he demeaned himself in an humble manner. In regard to the actual
change respecting his manifestations in heaven, or the withdrawing of
the symbols of his glory there, the Scriptures are nearly silent, and
conjecture is useless - perhaps improper. The language before us
fairly implies that he laid aside that which was expressive of his
being divine - that glory which is involved in the phrase "being in
the form of God" - and took upon himself another form and
manifestation in the condition of a servant.
In short, per the Barnes' notes, He did not lay aside any of His power or Godly characteristics, He merely did not exercise them.
Gills' commentary says this:
...He lost nothing of what he had; the glory of his divine nature was
covered, and out of sight;...
Vincent's Word Studies also agrees:
The general sense is that He divested Himself of that peculiar mode of
existence which was proper and peculiar to Him as one with God. He
laid aside the form of God. In so doing, He did not divest Himself of
His divine nature. The change was a change of state: the form of a
servant for the form of God. His personality continued the same. His
self-emptying was not self-extinction, nor was the divine Being
changed into a mere man. In His humanity He retained the consciousness
of deity, and in His incarnate state carried out the mind which
animated Him before His incarnation. He was not unable to assert
equality with God. He was able not to assert it.
as do Wesley's notes
Though he remained full, John 1:14, yet he appeared as if he had been
empty; for he veiled his fulness from the sight of men and angels.
Most of the commentaries that I can find agree with your original understanding of the phrase. He retained all His power and glory, he merely chose not to exercise it.
These, and a whole slew of other commentaries are available at http://bible.cc/philippians/2-7.htm