My 2 cents as an electrical engineer with a sad 25 years playing with computers:

Bad memory is commonly sold as good.
Bad connections are by far the absolute most common cause of 90% of all electronic failures.
(a) hard drive ribbon cables, with their push-on crimped connectors (swap out cables)
(b) bad pin connections on cables (unplug & replug cables)
(c) bad contacts on cards and memory sticks.
(d) heat problems, causing unpredictable behaviour
(e) static damage from unsafe handling of microcircuitry or bad grounds.
(f) electrical damage from improper connecting/disconnecting with power on.
(g) electrical damage from power on/off without proper cycle allowances.

Things you can do which often save hours, days or weeks of aggravation:

(1) Buy some pure contact cleaner (use only pure isopropyl alcohol not diluted with water like that sold in pharmacies)
and remove all grease and fingerprints on all contact surfaces.
This is a very common cause of memory and plug-in card faults.
Most people put their fingers all over gold-plated contact areas when installing new cards and memory strips.

(2) Double check all ribbon connections by carefully, firmly but gently removing and reconnecting/replacing them.
This also is a common cause of failure especially disk drives, as contact failure during writing or reading will destroy data on disk and in ram. Very often when one is installing or removing one part in a computer, another part is inadvertantly loosened or strained.

(3) Visually inspect all contact slots for damaged or loose or low-pressure contacts. (vga cards/memory sockets)
Typical scenarios are a machine is off, and cools down slightly, all contacts are good.
The machine is tested, all tests are passed.
The machine warms up slightly, and a connection somewhere fails, minutes, hours, or days later, on and off.

(4) Vibrations: Raise your computer off the floor or desk or shelf with about an inch of dense foam.
Do not bang desk, floor, shelf or case while hard-drives are on.
Noisey fans or supplies or moving parts that cause any vibrations should be replaced with quieter, less vibrating parts.
Do NOT scrimp on screws and fasteners that hold case parts or computer parts in machine.
These are important to ensure rigidity of the case and prevent failure of moving parts or failure to remove heat from critical components.

(5) Make sure key heat transfer surfaces are physically firmly and fully in contact.
make sure thermally conductive grease is applied where needed to ensure heat transfer. (this is often skipped with disastrous results)
Make sure all fans are connected, working properly, and that air flow is possible and not recycling the same air, building temperature.

Problems Cold Booting?

This is probably a bad connection, or a power supply problem, like weak capacitors which are taking too long to charge up and pull up the D.C. voltage to the boards and hard drives. Often with motors, (i.e.,hard drives) the current drain on a cold powersupply causes a voltage drop and the hardware then operates unreliably.
If you have a weak powersupply you can try to replace the caps yourself if you have the skills, or else just replace the whole unit, or remove some items from the computer to decrease the strain. Sometimes a powersuppy will have several connectors to power CDs, Floppies and Hard drives. Make sure that each hard drive is on a separate cable set coming from the supply box, as this should hopefully ensure that each drive is powered by a separate sub-section. Alternately, check that the CDROMs (which also draw current when they spin are on their own line, and that the floppy is powered from the same line as the CD/DVD, because usually these three units are not all running at once. When they are idle they should draw little or no power.
But if you have a couple of Hard Drives and a CD player on the same cabling, there is a chance that you are overloading one section of your power supply, but not efficiently using some other section.

Tools you'll need:

(a) A good work lamp or a bright flashlight. Don't rely on ambient light. Get in, light it up and have a good look.
(b) a medium and a small 'star' (Phillips) screwdrivers.
Extra handy is a snub-size medium screwdriver to get into cramped positions on the site, or between panels and cards.
(c) Needle-nosed pliers. Handy for removing and handling jumpers and cable connectors without damaging them.
(d) A can of air spray.
(e) A can of pure Isopropyl alcohol, and clean cotton balls.

(1) Shut off the computer. (don't skip this) And pull off a panel.

Phase I: Inspect and general clean

(2) Open up the computer and do a physical inspection. Look for buildup of dust around fans and contact areas, vents and cards. Look for cables that have been knocked or pulled and which are not properly plugged in. Check for moisture damage, electrocuted bugs or spiders, or (yes it happens) mouse droppings.

(3) Thoroughly clean out any dust and debris. To do this, you should have a can of compressed air you can pick up at any Radio Shack store, or hi-fi supply house.

(4) Carefully remove the ribbon cables one at a time, noting their orientation and re-install them. There is a red strip down one side of the ribbon to indicate which way they go. Usually the redline goes to the left, if you are facing the drive, from the front.

Phase 2: Get out your contact cleaner - pure isopropyl alcohol not diluted by water.

Any mechanical /removable electrical contacts should be inspected and cleaned. People often put fingerprints on gold contacts when installing cards and RAM. The result is later failure as salt, moisture and grease from fingers causes contacts to fail.

Warning: Don't overdo cleaning by dumping alcohol on motherboard, or attempting to wash components. Only use cleaner sparingly on contacts with a cotton ball. Never fire up or turn on power until all alcohol is completely evaporated and computer is thoroughly dry.

(5) Carefully remove the memory cards and reinstall them.
periodically, what happens is that there is a usually a bad connection on one of the un-soldered contacts in the computer which causes unusual behaviour or failure. unpredictible behaviour is almost always hardware related.

(6) Carefully remove PCI slot cards and reseat them, making sure they are securely held again by their holdin screws. Sometimes a card may not be exactly aligned, and this results in an unreliable connection.

Phase III: Test and Fan/powersupply checks

(7) Make sure the load on the power supply is as evenly distributed as possible. Put Hard drives on separate power leads, and put CDroms and Floppies on another lead. Use all the leads if possible, rather than putting your stuff all on one arm of the power supply.

(8) Reboot, noting whether all the fans in the unit are operating properly, especially any connected to the PROCESSOR or Video Card. If some fans aren't working, replace or seek some help at the store or repair shop.

Post the results if you are still getting errors.

Often with the cheap powersupplies, there is a flaw or a low power capability on one or two of the powersupply cables.
You do the right thing splitting up the power drain across different parts of the supply.
But I would look into upgrading quickly, before there is a power failure. A weak power supply is very unhealthy.
When you buy a new one, look for a 'quiet fan' , 'dual speed auto', or 'stealth' supply, and get one that is at least 100 watts more than you think you need. Never run a power supply at its max rating, especially if you leave it on 24/7. The extra few dollars you spend on a quieter supply by the way is worth its weight in gold for you as the person who has to listen to the noise all day.