Yes. There are components in the computer that can go bad - some go bad quicker than other components. Those components would be hard drives, power supplies, disk drives (CD, floppy...). Solid state devices (electronics) - the processor, the motherboard, the memory, the PCI cards - usually last a very long time. However, heat, static electricity and surges or brownouts can hurt these components.
It sounds like your processor is overheating for one. This is usually caused by dust buildup. Take the computer tower in the kitchen (or where there's no carpet) and open it up. Blow the dust out - depending on how bad it is, you might have to disassemble the computer to get it clean - dust can make its way anywhere. While you have the case open, it doesn't hurt to add an exhaust fan (no! not the kind in your bathroom ceiling!

) but it is a miniature fan that you can screw in your PC. Make sure it blows OUT and not in. On my computer, I have an intake fan and an exhaust fan - on the intake fan, I use a dryer sheet or two that I change out every four to six months. Since I have a central air cleaner, I don't have to worry much about dust.
The hard drive is about 11 years old. Hard drives have trouble making it past five years - especially if its used constantly and/or its not being treated well (laptop hard drives can attest to this) - it's time to get a new hard drive. You probably have no more than 5 GB - a reliable Seagate IDE hard drive that has an 80 GB capacity - 16 times the capacity of what you probably have - for $30.
It doesn't hurt to add memory as well. You can never have enough memory, but don't overdo it. Don't expect to put 4 GB in the machine - 512 MB of RAM is pushing it. If you are running anything below Vista - that's XP, ME, 2000 - you mentioned that you are running ME, 256 MB is great, even on XP.
Since you might want to replace the hard drive, just start from scratch and buy a copy of XP - bear in mind, however, XP won't be lightning fast on this system. You might have to scour and search hard for it- as XP was due off the shelves in June of this year. Or, you can save your 490 or so and get Linux, a great operating system, for free in many cases. Popular distributions include Ubuntu and Fedora - I use both and like the distributions really well. Bear in mind, though, that you can't expect a new release of these distributions to work well on your computer - you have an older processor and limited capabilities - so something like Fedora Core 4 would be okay. If you want a newer distribution, Xubuntu is the route you can take. Bear in mind, however, that Linux cannot run Windows programs without an emulator, and even with one, the software may not work.
Another option is to buy or build a computer - as long as you don't buy quad-core systems with a terabyte of hard disk space, you can have buy a computer for around $4 - 600, and it can be built for a lot cheaper.