I have a room I can enter from two different doors. Each has a rocker
type switch on the wall. The lights are flouresant tubes. Suddenly
one of the switches needed to always be in the down position in order
for the lights to have a chance of going on while using the other
switch. If this particular switch was not in the down position, then
the lights would always be off no matter what I do to the other
switch. I went to Home Depot and changed out both switches. It
fixed the problem completely for about 2-3 days.. Suddenly the same
problem is occurring again (now with the new rocker 3-way switches).
How could this be? Anyone have any idea what is going on here and
how to fix this problem?

Assuming the most common arrangement,

The way the switchs work is something like this:
---------==============----------*
Where the and are switches, and the
=== is a pair of travellers between the switches.

Normally, power comes in from the left, and switch
sends it to either the top or bottom traveller,
and switch either collects it from the same
traveller and sends it on to the light *.

From your description, one of the travellers
has either come loose at one end or the other,
or has an intermittent break in it, or one
of the switches is failing to make contact in
either the up or the down position.

If the light comes on with both
find a combination of position that makes
the light come on. Then flip both switches
once. The light should come on in that position
also. Since it doesn't, try rattling the
switch levers side to side. If the light
comes on, or the switch spits a spark,
then that's a bad switch.

If you get no results, take both switches apart,
after noting what wires go where, and check
each traveler for continuity. If you don't
have continuity, pray that there's a junction
box or something between the switches where
it's come apart. If you do have continuity,
replace both switches.