BOOT PROBLEMS, HARDWARE VIEW
This is a general outline as to why machines have boot problems.
GENERAL
Usually if the system tries to boot, i.e. you actually see the OS's OEM logo (Windows,etc.) it's usually a safe bet that the problem is software related. At that point the system, OS included, at least has access to the hard drive.
POWER SUPPLY UNIT or PSU
The PSU is designed to provide needed power to the hardware in any given system, it is NOT designed to prevent nor protect the system against electrical surges and/or spikes. If you own your own home you should have a spike arrestor installed, it's a relatively inexpensive device that only costs about $25 -$30. If you're not comfortable about how to install the device then have a qualified electrician do so, the cost should not be much more than the device itself. Secondly, or if you're in an apartment or you lease, get a computer grade power strip that also provides protection for your modem. Better yet, get a UPS (universal power system), that provides the function of a power strip, protects against surges/spikes, and provides emergency power to safely power down the system in case of a general power failure. Most of that should be a no brainer but it's often a simple step that is frequently overlooked.
Everything in and connected to the system depends on a fully functional PSU, if it fails or is simply overrun by a power surge/spike everything is open to damage. Such includes but is not limited to BIOS, CPU, RAM, etc.
A functional PSU fan can give a false positive as it operates on 12vDC.
As noted elsewhere one or more of the system board's power connectors HAS to be 12vAC hence the name switching power supply.
The system board usually must be grounded to the frame, don't assume that the ground cables from the PSU are doing the job adequately. Sometimes power switches located on the front of the machine (off the cable pigtail) have a ground lead that must be grounded to the frame.
BIOS
Short of the above PSU notations there's not much to go wrong with the BIOS. Various problems that arise with hardware usually are prompted by an improper setting and most likely at worst you'll suffer the consequences due inadequate device support from the BIOS. Without the BIOS everything short of the power supply is so much dead weight or scrap metal, etc.
CENTRAL PROCESSING UNIT OR CPU
With a defective CPU it's possible that the system will light up but will fall far short of completing POST. CPU problems arise from power spikes/surges and over heating.
RANDOM ACCESS MEMORY OR RAM
In nearly 15 years of trouble shooting systems I've yet to actually see defective RAM short of obvious physical damage. When tested on a testing machine, suspect RAM was proved to be functional.
HARD DISK AND FLOPPY DRIVES
Improper cable connections and the wrong jumper settings are the misused rule here. Where the ribbon cable is connected (upsided down) the system will give an error and probably will not boot, there's usually no permanent damage as applied voltage is going direct to ground resulting in no connection. Improper power cable connections are sometimes possible with a great deal of applied pressure, everything will be fine, that is until you power up the system, at that time your HD, etc. is DOA. The good news is that your data is still safely on the drive's platter(s) but getting access to them requires the use of data recovery technicians, and they aren't cheap, you really gotta want the data bad at that point. Better to backup, backup, backup, etc. and preferably to a removable media.
A defective BIOS will cause translation problems with the HD, etc.
DISCLOSURE
As noted above this is a general outline based upon 15 years of the "School of Hard Knocks". Trained technicians (hardware and software) usually aren't taught customer relations so their people communication skills fall short because their conversation is full "techno speak" that only confuses further the average end user and, as much as they might like to, they can't be there to help with your specific problem. Engineers usually have better people skills but usually access to them is limited or restricted. This can be a pain because when the machine has a problem it's usually after business hours or on the weekend when technical help is not readily available.
Hardware, new and used, does go bad, that however is the exception rather than the rule.
If you have a problem, please use the applicable forum subheading or your best guess.

RANDOM ACCESS MEMORY OR RAM
In nearly 15 years of trouble shooting systems I've yet to actually see defective RAM short of obvious physical damage. When tested on a testing machine, suspect RAM was proved to be functional.

I have been studying and repairing computers for 15 years myself,learned alot from trial and error also (The School of Hard Knocks).i am CompTIA certified and getting a Associates in Specialized Technology in PC Maintenance Technology.I do have to disagree on the RAM part.I have purchased RAM that was DOA on arrival and that was testing in more than one machine with no obvious physical damage.I have seen machines that didn't like certain brands of RAM like HP some i have worked on just wouldn't work with Kingston Value RAM and the RAM was rated for the system.Corsair I have yet to run across that problem.I have a stick of Ram now on my parts shelf that came out of a HP that i can't get to work in any HP that comes in.It is a good stick because i use it in my test machine.It is a Mega Trends stick with a HP sticker on it.

I have been studying and repairing computers for 15 years myself,learned alot from trial and error also (The School of Hard Knocks).i am CompTIA certified and getting a Associates in Specialized Technology in PC Maintenance Technology.I do have to disagree on the RAM part.I have purchased RAM that was DOA on arrival and that was testing in more than one machine with no obvious physical damage.I have seen machines that didn't like certain brands of RAM like HP some i have worked on just wouldn't work with Kingston Value RAM and the RAM was rated for the system.Corsair I have yet to run across that problem.I have a stick of Ram now on my parts shelf that came out of a HP that i can't get to work in any HP that comes in.It is a good stick because i use it in my test machine.It is a Mega Trends stick with a HP sticker on it.

First there's this:
"I've yet to actually see defective RAM short of obvious physical damage"
It's certainly nice to know of someone that does have some formal training in these boxes, however, as a formally trained technician in a entirely different field I can say that such is just a place to begin, the rest is by experience or "seat of the pants".
Although rare I do see equipment that is just as DOA as the RAM that you mentioned, but I tell the client that the best thing for them to do is take it back to the store if possible, best all the way around in the long run. With a lot of electronic parts, taking it back isn't an option even with the receipt, you take it out of the original packaging and it's YOURS.
In the past 5 yrs. or so there's been all sorts of RAM made for a variety of applications and machines, and some, even if they fit, don't work. When the x386 * x486 machines (and earlier) (Compaq and IBM RAM were also quite different) came out the basic difference (aside from speed and capacity) in RAM was parity and non-parity, you couldn't mix the two nor use either in a machine that wasn't designed for it.
There are, without doubt, a wide variety of mysteries that are run in to in any service arena that defy explanation.
I don't give suspected RAM problems much pause because it's never been much of an issue at the finish line. If there is one it's usually because someone stuck the wrong module in in the first place.
Ever see a glass Coke bottle with a Pepsi bottle cap?
A certain make of auto transmission, while sitting still, would go from Drive to Reverse for no explicable reason, oh yes, the engineers said it's impossible, unfortunately that meant beans to the TV cameraman that got it all on tape, and that ended up "News at 6 o'clock". Can you imagine the red faces at the designers conference table?
I had a machine that needed parts, the usual manner was used to order them, the manuf. wrote back that they didn't make any such thing. I peeled off the Mod. ID label and sent it and all the correspondence back that was received, yes, we made a mistake, and, oh btw, here's the parts you ordered.:)

You see alot of strange things after you have been doing this for a few years.that memory stick mentioned above by mega trends,well it is labeled as a 256.guess what,it has 128 chips on it.LOL

I am sure over your 15 years as me have done things with computers the manufacture said couldn't be done.but that is what keeps me interested in these crazy things all the things that make them up the new technology the things you can do with them.I only got the cert to just have it and getting the degree because i want to see why they make a big fuss about it.myself if i was gonna hire somebody in my own shop i would take experience over a degree.most people I know that actually knows computers are like you and me and have actually spent real time learning it instead of the way colleges teaches everything like it is straight forward.not everything is just cut and dry with these.I keep alot of notes as reference nobody can remeber all this stuff.

The prob. with the computer schools is that they only teach the student one aspect of the subject.
Service tech's are only taught how to repair.
Programmer's are only taught how to program.
Very few are actually taught how to operate them well.
You see where that's going.
To get the optimum out of these machines you have to have a reasonable amount of ALL aspects surrounding them.
For me it was either fix them myself or pay through the nose for it to be in and out of the shop all the time and my pocket would not allow the latter.
Experience does pay but it's sorely in short supply. Most big computer businesses actually would not hire someone like you and me, our minds can not be easily moulded to suit them. It would pay them in the long run but they don't see it that way. The only thing that matters to them is the short term "grab 4 the dough", looks really good on their quarterly report. And they wonder why they're at high risk and likely to go under quickly.

LOL your right on that.i just got my CompTIA cert last summer just because a buddy that has been in this for longer mentioned it.I thought what the hey.the degree i am doing a at home college because i am sure if i was in a class environment me and the professor would be butting heads alot.

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