No. Unless...
- The cat is healthy and more than ~seven-weeks old.
- If dropped from a height of more than ~1.5 feet and less than ~700 feet.
- The vertical velocity of the cat is less than ~60 mph (terminal velocity for cats).
A cat dropped upside-down turns over
very rapidly - in fact, it will do so
within its own standing height.
“How does a cat fall on its
feet?”, The New Scientist, vol. 7,
no. 189, pp. 1647-9.
Étienne-Jules Marey experimented on cats by shooting them (Video). With his chronophotographic gun.

The minimum height is a basic physics problem...
...use Newton's law of motion, which
says that mass times acceleration is
equal to the sum of all the forces
acting on the object. Since the cat is
falling only a short distance at a
fairly low velocity, it is safe to
assume that the only force acting on
the cat is gravity. Thus,
ma = -mg,
where m is the mass of the cat, a is
the acceleration of the cat, and -mg
is the force of gravity (assuming up
is positive), where g is a constant (g
= 979 cm/sec2 at a latitude like San Diego when you add centripetal
acceleration to the standard value
given for g, which is 980.7 cm/sec2 ).
See: Differential Equations and
Integration for all the fun math.
The maximum height and velocity are related. Once weightless the cat has no "down" and might not maintain proper orientation.
When a cat, blindfolded and lying on
its back, is dropped, it immediately
turns into normal position to land on
its feet, indicating that a visual cue
is not involved in this spatial
orientation. Function of such
orientation response of cats was
examined during microgravity produced
by aircraft parabolic flights. The
three-week-old kittens whose postural
righting reflex was not developed,
floated upside-down in air during
weightlessness, whereas in the eight-
and twelve-week-old ones whose
reflexes were well established, the
reflex acted for the initial several
seconds but ceased thereafter during
the weightless state, with or without
blindfold;
Disorientation
of animals in microgravity.
However this might not be a bad thing...
What appears to happen is that in
falls from above 7 stories, cats have
time to reach terminal velocity, the
speed at which their drag due to wind
resistance equals their weight due to
gravity and they stop accelerating. At
this speed, the cat relaxes its limbs
and is better able to survive the
impact (at a slight cost – they get
less broken limbs, but more chest
injuries).
source
Finally we have this excellent interesting questionable data from the Annals of Improbable Research...
Cats have excellent balance, and are
remarkably acrobatic. When turned
upside down and dropped from a height,
a cat generally has the ability to
land on its feet. Until now, no one
has systematically investigated the
limits of this phenomenon. In this
study, I dropped a cat upside down
from various heights, and observed
whether the cat landed on its feet.

Does a cat land on its feet when
dropped from a height of less than one
foot? This preliminary study indicates
that the answer may be no. However,
further experiments, preferably with
the same cat, are needed to settle the
question.
Does a Cat Always Land on
Its Feet ? Fiorella Gambale,
Ph.D.
NOTE: A cat with a piece of buttered toast attached to its back, butter side up, will not land on its feet. In fact the forces create by this configuration will cause the cat to hover.
Related cat physics: Cat Swinging. Principles & Mechanics.