Answer
This question was answered on July 14, 1999.
Much of the information contained herein may have changed since posting.
It is assumed that you have an ATX motherboard, and it is recommended that you have just one power supply, because if you install the second power supply it would not be connected to the motherboard.
ATX has a "Soft Power" Support where the ATX power supply is turned on and off using signalling from the motherboard, not a physical toggle switch. This allows the PC to be turned on and off under software control, allowing much improved power management. For example, with an ATX system you can configure Windows 95 so that it will actually turn the PC off when you tell it to shut down.
With the second power supply this feature "Soft power support", is not possible as this would not be connected to the motherboard. Therefore this is not recommended.
If you run out of drive power connectors, you can usually get another one by using a "Y adapter" or "Y splitter". Named because of its shape, the adapter splits a single power connector into two, letting you use two devices. Of course, this assumes that your supply has enough power to drive the additional device.
Warning: There have been reports of incorrectly wired Y-adapters. Watch out for them; they have the potential to damage your equipment. It's pretty easy to see if the adapter has been wired correctly by inspecting it carefully.