Not necessarily. Most wall adaptors are not regulated, this means that the voltage they produce varies with load. For a "5V" adaptor with minimal load the voltage may be as high as 12 or 15 V.
For this reason, most equipment designed to run off cheap wall adaptors contain internal voltage regulators to protect the sensitive electronics.
If you can be sure that the original PSU was unregulated, then yes a 6V or 9V adaptor will work (as long as the polarity is right and the PSU is powerful enough - mA high enough). The problem is that the higher voltage produced by the PSU will mean more work for the voltage regulator - as the heatsinking for the regulator was designed for the original PSU, it may overheat.
If the PSU is a higher voltage and a lower current rating, then you can overheat or burn the PSU.
I've got a cheap network hub which takes 7V - internally there is a 5V regulator. I lost the original PSU and ran it for a bit from a 12V PSU - it worked fine, but the voltage regulator would get so hot that even the case of the hub got painfully hot to touch.